How do you factor the expression 4x^2 + 16x + 15?

Dec 30, 2015

$4 {x}^{2} + 16 x + 15 = \left(2 x + 3\right) \left(2 x + 5\right)$

Explanation:

I'm going to explain this using the most common method of factorising: by splitting the middle term.

The first step is to multiply the coefficient of ${x}^{2}$ with the constant. We get:

$4 \cdot 15 = 60$

Now, we need to find the pair of factors of $60$ whose sum or difference will give us the coefficient of $x$, i.e., $16$.

$60$ has the following pairs of factors:

$\left(1 , 60\right) , \left(2 , 30\right) , \left(3 , 20\right) , \left(5 , 12\right) , \left(6 , 10\right)$

With a quick glance, it's clear that the sum of the factors in the pair $\left(6 , 10\right)$ is $16$.

Great! So now we split the coefficient of middle term $\left(16\right)$ as a sum of $6$ and $10$ as:

$4 {x}^{2} + \left(6 + 10\right) x + 15$
$4 {x}^{2} + 6 x + 10 x + 15$

Note: It doesn't matter if you reverse the order and split $16 x$ as $10 x + 6 x$, you'll get the same result!

Now, we must take out common factors from the first two terms and then the next two terms:

$2 x \left(2 x + 3\right) + 5 \left(2 x + 3\right)$

Now, we can take $\left(2 x + 3\right)$ to be common, to get:

$\left(2 x + 3\right) \left(2 x + 5\right)$

and voila, that's the factored expression!