3.43: Polynomial factors then exponential equation

The polynomial $p$ is such that $p(x) = 3x^{3} - 7x^{2} + ax + b$, where $a$ and $b$ are integers. It is given that $p'(-1) = 21$ and that $x - 2$ is a factor of $p(x)$. Find the values of $a$ and $b$.

Hence write $p(x)$ as a product of linear factors with integer coefficients.

Using your values of $a$ and $b$, solve the equation $3\mathrm{e}^{6y} - 7\mathrm{e}^{4y} + a\mathrm{e}^{2y} + b = 0$.