Answer
$a=3$ and $b=-3/2$
Work Step by Step
Step 1. Based on the given piecewise function, $f′(x)=a,x\gt-1$ and $f′(x)=2bx,x\leq-1$.
Step 2. For the function to be differentiable for all x-values, the limits of the above two derivatives need to be equal when $x\to-1$, thus $ \lim_{x\to-1^+}(a)=\lim_{x\to-1^-} (2bx)$ which gives $a=2b(-1)$ or $a=-2b$
Step 3. The function needs to be continuous at $x=-1$, thus $ \lim_{x\to-1^+}(ax+b)=\lim_{x\to-1^-} (bx^2-3)$ which gives $-a+b=b-3$ or $a=3$ and $b=-a/2=-3/2$
