factoring a trinomial
A trinomial like x^2 + 5x + 6 often hides two binomials inside it. Factoring the trinomial means finding those binomials — here (x + 2)(x + 3) — so that multiplying them back out gives the trinomial you started with. It is a guess-and-check guided by arithmetic, not pure luck.
For a simple trinomial x^2 + bx + c (leading coefficient 1), you look for two numbers that multiply to c and add to b. For x^2 + 5x + 6 you need two numbers with product 6 and sum 5: that is 2 and 3, giving (x + 2)(x + 3). When the leading coefficient is not 1, such as 2x^2 + 7x + 3, the search is harder and people often turn to the AC method or reverse FOIL.
Be honest about the limits: many trinomials do not factor into binomials with whole-number (or even rational) entries. The product-and-sum search may simply fail to turn up a valid pair, in which case the trinomial is prime over the rationals and you must reach for the quadratic formula or completing the square instead.
x^2 - 7x + 12 = (x - 3)(x - 4), because (-3)(-4) = 12 and (-3) + (-4) = -7.
Product 12, sum -7 — both numbers are negative.
For x^2 + bx + c: if c is positive, the two numbers share the sign of b; if c is negative, they have opposite signs. This sign rule narrows the search before you even start.