3.67: Binomial expansion in descending powers

Find the rational numbers $a$, $b$ and $c$ such that the first three terms, in descending powers of $x$, in the expansion of $\left(3x^{2} - \dfrac{1}{2x}\right)^{5}$ can be written as $ax^{10} + bx^{7} + cx^{4}$.

Hence find the coefficient of $x^{4}$ in the expansion of $\left(3x^{2} - \dfrac{1}{2x}\right)^{5}\left(1 + \dfrac{1}{x^{3}}\right)^{2}$.