completely factored

A polynomial is completely factored when none of its factors can be broken down any further. It is the factoring equivalent of reducing a fraction to lowest terms: you keep going until there is nothing left to simplify. Stopping early is a common reason to lose marks, because a 'factored' answer that still contains a factorable piece is not finished.

Reaching complete factorization usually means applying several techniques in turn. First pull out the greatest common factor; then look for a special pattern (difference of squares, sum or difference of cubes, perfect-square trinomial); then try trinomial factoring or grouping; and at each new factor, ask again whether it factors. You are done only when every remaining factor is prime over your number system.

Watch especially for factors hidden inside other factors. For example, x^4 - 16 looks like a difference of squares giving (x^2 + 4)(x^2 - 4), but the second factor is itself a difference of squares, so the complete form is (x^2 + 4)(x + 2)(x - 2). Stopping at the first step would be only partly factored.