2.14: Binomial product: find $a$, $b$ and $c$

The first four terms in the expansion of $(1 + ax)^{5}(2 + bx)$ are $2 + 32x + 210x^{2} + cx^{3}$, where $a$, $b$ and $c$ are integers. Show that $3a^{2} - 16a + 21 = 0$ and hence find the values of $a$, $b$ and $c$.