Answer
$a=\left\{ -2,6 \right\}$
Work Step by Step
Expressing the given equation in the form $ax^2+bx+c=0,$ the given equation is equivalent to
\begin{array}{l}\require{cancel}
(a+1)(a-5)=7
\\\\
a(a)+a(-5)+1(a)+1(-5)=7
\\\\
a^2-5a+a-5=7
\\\\
a^2-5a+a-5-7=0
\\\\
a^2-4a-12=0
.\end{array}
Using the factoring of trinomials in the form $x^2+bx+c,$ the $\text{
equation
}$
\begin{array}{l}\require{cancel}
a^2-4a-12=0
\end{array} has $c=
-12
$ and $b=
-4
.$
The two numbers with a product of $c$ and a sum of $b$ are $\left\{
-6,2
\right\}.$ Using these two numbers, the $\text{
equation
}$ above is equivalent to
\begin{array}{l}\require{cancel}
(a-6)(a+2)=0
.\end{array}
Equating each factor to zero (Zero Product Property), and then isolating the variable, the solutions are $
a=\left\{ -2,6 \right\}
.$
