Answer
$b=-\displaystyle \frac{1}{2}$
Work Step by Step
The piecewise defined functions are continuous on their intervals because they are polynomials.
The only problem that could arise is at $x=-2$.
Find the one-sided limits at $x=-2$ and set them to be equal, in which case there will exist a limit and g will be continuous at $x=-2$:
$\displaystyle \lim_{x\rightarrow-2^{-}} g(x)=\displaystyle \lim_{x\rightarrow-2^{-}}x=-2$
$\displaystyle \lim_{x\rightarrow-2^{+}} g(x)=\displaystyle \lim_{x\rightarrow-2^{+}}bx^{2}=4b$
Set the one-sided limits to be equal
$4b=-2$
$b=-\displaystyle \frac{1}{2}$
