Question 207·Hard·Equivalent Expressions

For all real numbers $x \neq -3$, the expression

can be rewritten in the form

where $p$, $q$, $r$, and $s$ are integers.

If $x = 1$ is a zero of the polynomial $x^3 + p x^2 + q x + 13$, what is the value of $p + q$?