Align enumerate to text and items to numbering for exercises and answers

Question

Problems:

1. I wish to align the items to the text for the exercises.
2. I wish to move the a) so it is directly to the right of the number 1 for answers.

Minimal Working Example (MWE):

\documentclass{article}
\usepackage{amsmath}
\usepackage[shortlabels,inline]{enumitem}
\usepackage[lastexercise,answerdelayed,exercisedelayed]{exercise}
\usepackage{showframe}

% --------------------------------------------------------------
% Exercise and answer options
% --------------------------------------------------------------
\renewcommand{\ExerciseName}{}
\renewcommand{\AnswerName}{}
\renewcommand{\ExerciseHeader}{\noindent{\textbf{\ExerciseHeaderNB}} \ExerciseTitle}
\renewcommand{\AnswerHeader}{\noindent{\textbf{\ExerciseHeaderNB}} \ExerciseTitle}
\setlength{\ExerciseSkipBefore}{0pt}

\begin{document}

% --------------------------------------------------------------
% Exercise 1
% --------------------------------------------------------------
\begin{Exercise}
Some text here \\
\begin{enumerate}[\bfseries\alph*)]
  \item $x^{2}+2 x-8$
  \item $x^{2}+2 x+8$
  \item $2 x^{2}+x$
  \item $-3+4 x-x^{2}$
\end{enumerate}
\end{Exercise}

% --------------------------------------------------------------
% Answer 1
% --------------------------------------------------------------
\begin{Answer}
\begin{enumerate}[\bfseries\alph*)]
  \item $(x+1)^{2}-9,(x-2)(x+4)$
  \item $(x+1)^{2}+7$
  \item $2\left(x+\frac{1}{4}\right)^{2}-\frac{1}{8}, 2 x\left(x+\frac{1}{2}\right)$
  \item $1-(x-2)^{2},-(x-1)(x-3)$
\end{enumerate}
\end{Answer}

% --------------------------------------------------------------
% Prints out exercises and answers
% --------------------------------------------------------------
\section{Exercises}
\shipoutExercise
\section{Answers}
\shipoutAnswer

\end{document}

Current output:

Desired output:

To have the items align to the text on the left for the exercises and let the enumerate list begin directly to the right of the answer number and then keep them aligned.

Answer 1

If I've well understood what you want, here is a solution. I've taken the liberty to add a small space after the comma in some answers.

\documentclass{article}
\usepackage{amsmath}
\usepackage[shortlabels,inline]{enumitem}
\usepackage[lastexercise,answerdelayed,exercisedelayed]{exercise}
\usepackage{showframe}

% --------------------------------------------------------------
% Exercise and answer options
% --------------------------------------------------------------
\renewcommand{\ExerciseName}{}
\renewcommand{\AnswerName}{}
\renewcommand{\ExerciseHeader}{\noindent{\textbf{\ExerciseHeaderNB}} \ExerciseTitle}
\renewcommand{\AnswerHeader}{\noindent{\textbf{\ExerciseHeaderNB}} \ExerciseTitle}
\setlength{\ExerciseSkipBefore}{0pt}

\begin{document}

% --------------------------------------------------------------
% Exercise 1
% --------------------------------------------------------------
\begin{Exercise}
Some text here
\begin{enumerate}[\bfseries\alph*), wide=0pt]
  \item $x^{2}+2 x-8$
  \item $x^{2}+2 x+8$
  \item $2 x^{2}+x$
  \item $-3+4 x-x^{2}$
\end{enumerate}
\end{Exercise}

% --------------------------------------------------------------
% Answer 1
% --------------------------------------------------------------
\begin{Answer}
\begin{enumerate}[\bfseries\alph*), topsep=0pt, before= \vspace*{-\baselineskip}]
  \item $(x+1)^{2}-9,\:(x-2)(x+4)$
  \item $(x+1)^{2}+7$
  \item $2\left(x+\frac{1}{4}\right)^{2}-\frac{1}{8},\: 2 x\left(x+\frac{1}{2}\right)$
  \item $1-(x-2)^{2},\:-(x-1)(x-3)$
\end{enumerate}
\end{Answer}

% --------------------------------------------------------------
% Prints out exercises and answers
% --------------------------------------------------------------
\section{Exercises}
\shipoutExercise
\section{Answers}
\shipoutAnswer

\end{document} 

Update: Another alignment for the questions in the exercise part, and a small improvement of the Answers part:

\documentclass{article}
\usepackage{amsmath}
\usepackage[shortlabels,inline]{enumitem}
\usepackage[lastexercise,answerdelayed,exercisedelayed]{exercise}
\usepackage{showframe}
% --------------------------------------------------------------
% Exercise and answer options
% --------------------------------------------------------------
\renewcommand{\ExerciseName}{}
\renewcommand{\AnswerName}{}
\renewcommand{\ExerciseHeader}{\noindent{\textbf{\ExerciseHeaderNB}} \ExerciseTitle}
\renewcommand{\AnswerHeader}{\noindent{\textbf{\ExerciseHeaderNB}} \ExerciseTitle}
\setlength{\ExerciseSkipBefore}{0pt}
\usepackage{xpatch} 
\renewcommand{\QuestionNB}{\textbf{\alph{Question})\ }}
\setlength{\QuestionBefore}{0.67em}

\begin{document}

% --------------------------------------------------------------
% Exercise 1
% --------------------------------------------------------------
\begin{Exercise}
 Some text here
 \Question $x^{2}+2 x-8$
 \Question $x^{2}+2 x+8$
 \Question $2 x^{2}+x$
 \Question $-3+4 x-x^{2}$
\end{Exercise}

% --------------------------------------------------------------
% Answer 1
% --------------------------------------------------------------
\begin{Answer}
\begin{enumerate}[\alph*), font=\bfseries, topsep=0pt, leftmargin=2.8em, before= \vspace*{-\baselineskip}]
  \item $(x+1)^{2}-9,\:(x-2)(x+4)$
  \item $(x+1)^{2}+7$
  \item $2\left(x+\frac{1}{4}\right)^{2}-\frac{1}{8},\: 2 x\left(x+\frac{1}{2}\right)$
  \item $1-(x-2)^{2},\:-(x-1)(x-3)$
\end{enumerate}
\end{Answer}

% --------------------------------------------------------------
% Prints out exercises and answers
% --------------------------------------------------------------
\section{Exercises}
\shipoutExercise
\section{Answers}
\shipoutAnswer

\end{document} 

To have the exercises and the answers in two columns, you can use the tasks package:

\documentclass{article}
\usepackage{amsmath}
\usepackage[lastexercise,answerdelayed,exercisedelayed]{exercise}
\usepackage{tasks}
\usepackage{showframe}
% --------------------------------------------------------------
% Exercise and answer options
% --------------------------------------------------------------
\renewcommand{\ExerciseName}{}
\renewcommand{\AnswerName}{}
\renewcommand{\ExerciseHeader}{\noindent{\textbf{\ExerciseHeaderNB}} \ExerciseTitle}
\renewcommand{\AnswerHeader}{\noindent{\textbf{\ExerciseHeaderNB}} \ExerciseTitle}
\setlength{\ExerciseSkipBefore}{0pt}

\begin{document}

% --------------------------------------------------------------
% Exercise 1
% --------------------------------------------------------------
\begin{Exercise}
 Some text here
\begin{tasks}[label-format=\bfseries](2)
 \task $x^{2}+2 x-8$
 \task $x^{2}+2 x+8$
 \task $2 x^{2}+x$
 \task $-3+4 x-x^{2}$
\end{tasks}
\end{Exercise}

% --------------------------------------------------------------
% Answer 1
% --------------------------------------------------------------
\begin{Answer}\vspace*{-\dimexpr\baselineskip + \topsep}
\begin{tasks}[label-format=\bfseries](2)
\task $x^{2}+2 x-8$
\task $x^{2}+2 x+8$
      \task $2\left(x+\frac{1}{4}\right)^{2}-\frac{1}{8},\: 2 x\left(x+\frac{1}{2}\right)$
      \task $1-(x-2)^{2},\:-(x-1)(x-3)$
\end{tasks}
\end{Answer}

% --------------------------------------------------------------
% Prints out exercises and answers
% --------------------------------------------------------------
\section{Exercises}
\shipoutExercise
\section{Answers}
\shipoutAnswer

\end{document}