A Distorting Problem

Without loss of generality, we can assume the coefficient of $x^7$ in $Q(x)$ is $1$ . So, let $Q(x)=x^7+R(x)$ , where $R(x)$ is a polynomial of degree less than $7$ .

Differentiating,

$\begin{aligned} P(x)\cdot e^{kx}&=\frac{d}{dx} \left[ \left( x^7+R(x) \right)e^{4x}\right] \\ &=\left( 4x^7+4R(x)+7x^6+R'(x) \right) e^{4x}\end{aligned}$

Comparing the two sides, we immediately see that $n=7$ and $k=4$ .

The ratio in the limit is just the ratio of the leading coefficients of $P$ and $Q$ (since they have the same degree), which is $4$ .

Putting this together, the answer is $7+7+4+4=\boxed{22}$ .