Factorize $4x^4 - 27x^2 - 81$:

As a quadratic in $x^2$, $4x^4-27x^2-81=(4x^2+9)(x^2-9)$ since $4\times(-81)=-324$ splits as $-36$ and $+9$. The factor $x^2-9$ is a difference of squares, giving $(4x^2+9)(x-3)(x+3)$.