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The domain of every polynomial is all real numbers. The sign of $a$ is negative so the function has a maximum value at the vertex. This gives the range as $(- \infty, - 7]$

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L A T E X

\begin{question}Find the domain and range of $f(x)=- 5 \left(x + 6\right)^{2} - 7 $
    \soln{9cm}{The domain of every polynomial is all real numbers. The sign of $a$ is negative so the function has a maximum value at the vertex.  This gives the range as $(-\infty,-7]$}

\end{question}

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L A T E X

\documentclass{article}
\usepackage{tikz}
\usepackage{amsmath}
\usepackage[margin=2cm]{geometry}
\usepackage{tcolorbox}

\newcounter{ExamNumber}
\newcounter{questioncount}
\stepcounter{questioncount}

\newenvironment{question}{{\noindent\bfseries Question \arabic{questioncount}.}}{\stepcounter{questioncount}}
\renewcommand{\labelenumi}{{\bfseries (\alph{enumi})}}

\newif\ifShowSolution
\newcommand{\soln}[2]{%
\ifShowSolution%
\noindent\begin{tcolorbox}[colframe=blue,title=Solution]#2\end{tcolorbox}\else%
\vspace{#1}%
\fi%
}%
\newcommand{\hideifShowSolution}[1]{%
\ifShowSolution%
%
\else%
#1%
\fi%
}%
\everymath{\displaystyle}
\ShowSolutiontrue

\begin{document}\begin{question}(10pts) The question goes here!
    \soln{9cm}{The solution goes here.}

\end{question}\end{document}

Find the domain and range of <img class="equation_image" title=" \displaystyle f(x)=- 5 \left(x + 6\right)^{2} - 7 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%3D-%205%20%5Cleft%28x%20%2B%206%5Cright%29%5E%7B2%7D%20-%207%20%20" alt="LaTeX: \displaystyle f(x)=- 5 \left(x + 6\right)^{2} - 7 " data-equation-content=" \displaystyle f(x)=- 5 \left(x + 6\right)^{2} - 7 " />

The domain of every polynomial is all real numbers. The sign of <img class="equation_image" title=" \displaystyle a " src="/equation_images/%20%5Cdisplaystyle%20a%20" alt="LaTeX: \displaystyle a " data-equation-content=" \displaystyle a " /> is negative so the function has a maximum value at the vertex. This gives the range as <img class="equation_image" title=" \displaystyle (-\infty,-7] " src="/equation_images/%20%5Cdisplaystyle%20%28-%5Cinfty%2C-7%5D%20" alt="LaTeX: \displaystyle (-\infty,-7] " data-equation-content=" \displaystyle (-\infty,-7] " />