Q.Factorize: $x^4+1$. (Hint: Try completing the square.)

A library of standard identities (valid for all real $x,a,b$, and $n\in N$ where noted) is the main tool for factoring polynomials and constructing them from given conditions:

Completing the square to factor. A quartic like $x^4+1$ has no rational linear/quadratic factor directly, but adding and subtracting $2x^2$ turns it into a difference of squares $(x^2+1)^2-(\sqrt2x)^2$, which then factors normally.

Method of undetermined coefficients. To construct a polynomial from given zeros and/or function values: write it with unknown coefficients (or, if the zeros are already known, in factored form with one unknown overall scale constant), then use 'equal polynomials have equal same-power coefficients' (or substitute the given conditions directly) to solve for the unknowns. This same coefficient-matching idea proves polynomial-divisibility identities (matching a quotient's unknown coefficients) and derives closed forms for sums like $1+2+\cdots+n=\dfrac{n(n+1)}2$ by positing a polynomial form for the sum and matching how it changes from $n$ to $n+1$.