Factoring is turning an expression like 4x² - 10x - 6 into the product of simpler expressions, such as 2(x - 3)(2x + 1).

To do this, you must first isolate the greatest common factor. Oftentimes it is 1. In my example, the greatest common factor is 2. All the terms can be divided by 2. The new expression is 2(2x² - 5x - 3). You can now ignore the common factor as long as you remember to add include it when you finish. Let's factor 2x² - 5x - 3.

Factoring trinomials is done by finding two binomials (ax - b) that multiply to your trinomial. You could guess at any values of a and b for both binomials, but this would take a long time, and the trinomial might not even be factorable. Instead, you can use patterns.

The b values of both binomials always, always, always multiply to the trinomial's last value, the constant. You can use FOIL to observe this. That means that if the constant is 10, the only possible integer values for the binomials' constants are 2 and 5, one each.

The a values are the same, but with the coefficient of the x² term. If the polynomial is 2x² + 9x + 4, then you can only have ±2 and ±1 as possible integer values of a. If the expression is x² - 5x + 4, then the coeffients in the binomial are only 1 (assuming only integers).

Now, just try combinations (permutations?) of the a and b values in the binomials until you get something that works. To test it, use FOIL. If you practice, you'll notice patterns to solve it more easily. For example, when the trinomial's coefficient of x² is 1, the two b values of the binomials must add or subtract to the trinomial's coefficient of x. If you've tried all the possible factors (don't forget negatives!), then the trinomial is unfactorable.

Now multiply your factored trinomial by the common factor from Step 1.

Now, if you're factoring polynomials of powers more than 2, you'll have to do something more straightforward, but harder. I'll explain it if asked.

One more thing: if you're finding x in a trinomial that equals zero (e.g. ax² + bx + c = 0), you can just use the quadratic formula (with the added bonus that you can find x even if it's unfactorable): x=(-b ± √(b² - 4ac))/(2a). Wait! when using the quadratic formula, the trinomial is assumed to be slightly different in a way that makes it possible for some values to be extraneous (but it won't miss values). Test both values given by the quadratic formula to ensure that they do, in fact make it equal zero.