Answer
$a.\quad \left\{\begin{array}{l}
x\neq-5\\
x\neq 5
\end{array}\right. $
$ b.\quad$ Solution set = $\{7\}$
Work Step by Step
Factor the denominators:
$\displaystyle \frac{4}{x+5}+\frac{2}{x-5}=\frac{32}{(x+5)(x-5)}$
$a.$
The denominators in the equation must not be zero:
$\left\{\begin{array}{l}
x+5\neq 0\\
x-5\neq 0\\
(x+5)(x-5)\neq 0
\end{array}\right\}\Rightarrow\left\{\begin{array}{l}
x\neq-5\\
x\neq 5
\end{array}\right. \qquad(*)$
$b.$
Multiply both sides with the LCD,$\quad (x+5)(x-5)$
$(x+5)(x-5) \displaystyle \left[ \frac{4}{x+5}+\frac{2}{x-5} \right]= (x+5)(x-5) \left[\frac{32}{(x+5)(x-5)} \right]\quad$ ... distribute and simplify
$ 4(x-5)+2(x+5)=32\quad$ ... distribute and simplify
$4x-20+2x+10=32$
$ 6x-10=32\quad$ ... add $10$ to both sides
$6x=42$
$ x=7 \quad$ ... satisfies (*), a valid solution.
Solution set = $\{7\}$
