Answer
$a=-3$
Work Step by Step
Step 1. Based on the given piecewise function, $g'(x)=a, x\lt0$ and $g'(x)=2x-3, x\geq0$.
Step 2. For the function to be differentiable for all x-values, the limits of the above two derivatives need to be equal when $x\to0$, thus $\lim_{x\to0^-}a=\lim_{x\to0^+}(2x-3)$ which gives $a=-3$
