Factoring by Substitution — Cheat sheet

Key method

For $x^4+bx^2+c$, put $u=x^2$. The expression becomes $u^2+bu+c$. When solving over the reals, remember that $u=x^2\ge0$.

Example

Factor $x^4-5x^2+4$. With $u=x^2$,

Substitute back to get $(x^2-1)(x^2-4)$, then factor both differences of squares:

Avoid this mistake

Do not stop a solution at $u=1,4$ when the question asks for $x$. Each positive square value gives two real roots.