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To find possible rational roots of a polynomial with integer coefficients, list all factors of the constant term $a_0$ over factors of the leading coefficient $a_n$. These are the potential rational zeros.

For quadratics in the form $ax^2 + bx + c$, multiply $a$ and $c$. Find two numbers that multiply to $ac$ and add to $b$. Use these to split the $bx$ term and then factor by grouping.

Group terms with common factors and factor out the greatest common factor from each group.

Use the identity $a^2 - b^2 = (a + b)(a - b)$ to factor a binomial that is a difference of two squares.

Use the formulas $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ and $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ to factor the sum or difference of two cubes.

Rewrite the quadratic in the form $(x + p)^2 = q$ and solve for $x$.

Factor out the largest factor that is common to all terms in the polynomial.

Use the identities $(a + b)^2 = a^2 + 2ab + b^2$ and $(a - b)^2 = a^2 - 2ab + b^2$ to factor trinomials that are perfect squares.

For a polynomial of the form $a_nx^n + a_{n-1}x^{n-1} + ... + a_0$, use Eisenstein's Criterion to determine if it is irreducible over the integers.

For quadratics where $a=1$, find two numbers that multiply to $c$ and add to $b$. For $a \ne 1$, use the AC method or other techniques like completing the square.

For harder quadratics or when other methods fail, use the quadratic formula $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ to find the roots, then express as factors.