jmc

We can use the coefficient of the $x^3$ term to find $b$. On the right we have $-14x^3$, and on the left, the only $x^3$ terms we will get when we expand are $-3x(a{x}^2)$ and $5x^2(bx)$.

So we must have $$-3a{x}^3+5b{x}^3=-14x^3$$which means $$5b-3a=-14$$To find $a$, we use the same reasoning and look at the $x^4$ terms. On the right we have $15x^4$, and on the left, the only $x^4$ term we will get when we expand is $5x^2(a{x}^2)$. Then we know that $$5a{x}^4=15x^4$$which means that $a=3$.

Then $5b-3(3)=-14$ and $b=\boxed{-1}$.