Fermat 2004 Q24

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Problem Statement

The polynomial $f(x)$ satisfies the equation $f(x) - f(x - 2) = (2x - 1)^2 \;$ for all $x$. If $p$ and $q$ are the coefficients of $x^2$ and $x$, respectively, in $f(x)$, then $p + q$ equals

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Solution

$$f(x) - f(x - 2) = (2x - 1)^2 \;\; \forall x$$ $$x = \dfrac{1}{2}, \implies f(\dfrac{1}{2}) = f(\dfrac{-3}{2})$$