AC method

When a trinomial ax^2 + bx + c has a leading coefficient a that is not 1, the simple product-and-sum trick stops working directly. The AC method gives you a reliable recipe: instead of factoring c, you work with the product a times c, and that single change turns a hard trinomial into a four-term polynomial you can handle by grouping.

The steps: compute the product a·c (this is the 'AC'); find two numbers whose product is a·c and whose sum is the middle coefficient b; split the middle term bx into those two pieces; then factor the resulting four terms by grouping. For 2x^2 + 7x + 3, a·c = 6, and 6 and 1 multiply to 6 and add to 7, so 2x^2 + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (x + 3)(2x + 1).

The AC method is systematic rather than inspired, which is its virtue: it tells you exactly what two numbers to hunt for. Its limit is the same as all rational factoring — if no integer pair multiplies to a·c and adds to b, the trinomial does not factor over the integers, and the method honestly reports failure.