Answer
See below.
Work Step by Step
a) $f(g(x))=f(2x+5)=\frac{2x+5+2}{2x+5-2}=\frac{2x+7}{2x+3}$. The denominator cannot be $0$, which is when $2x+3=0\\3=-2x\\x=-1.5$.
Thus the domain is all real numbers apart from $-1.5$.
b) $g(f(-2))=g(\frac{-2+2}{-2-2})=g(0)=2\cdot0+5=5$
c) $f(g(-2))=\frac{2(-2)+7}{2(-2)+3}=\frac{3}{-1}=-3$
