How Do I Factor This Expression?

The following question is adapted from one asked by user "abel muroi" on 1-29-2015. The user asked:

Question:

How can i factor \(f(x) = 2x^2 + x - 6\)?

I'm trying to convert the function into the standard form, but I need help.

Can anyone tell me how to factor this?

Answer:

Quick note on terminology: \(2x^2 + x - 6\) is already in standard form. What you want is the factored form, which means rewriting it as a product of two binomials like \((x + \text{something})(x + \text{something else})\).

When the leading coefficient is 1, factoring is pretty straightforward. But here the leading coefficient is 2, which makes it a little trickier. The method that works reliably is called factoring by grouping. Here's how it goes.

First, multiply the leading coefficient by the constant: \(2 \times (-6) = -12\). Now find two numbers that multiply to \(-12\) and add to \(1\) (the middle coefficient). A little trial and error gives us 4 and -3, since \(4 \times (-3) = -12\) and \(4 + (-3) = 1\).

Use those two numbers to split the middle term, then factor by grouping:

$$ 2x^2 + x - 6 $$ $$ 2x^2 + 4x - 3x - 6 $$ $$ 2x(x + 2) - 3(x + 2) $$ $$ (2x - 3)(x + 2) $$

So the factored form is \(f(x) = (2x - 3)(x + 2)\).

You can always check your answer by multiplying it back out using FOIL. If you get the original expression, you factored it correctly.