# 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!
{"ft_lang_label":"__label__en","ft_lang_prob":0.877327,"math_prob":1.000002,"size":652,"snap":"2020-10-2020-16","text_gpt3_token_len":157,"char_repetition_ratio":0.125,"word_repetition_ratio":0.0,"special_character_ratio":0.2285276,"punctuation_ratio":0.09774436,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999378,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-02-19T20:13:10Z\",\"WARC-Record-ID\":\"<urn:uuid:d6becb65-c228-45bb-8e4a-3d443857abae>\",\"Content-Length\":\"35420\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0d39d784-af5c-4ac4-b862-c256f6a972ea>\",\"WARC-Concurrent-To\":\"<urn:uuid:54732b1a-0e66-4027-8fca-5ceab00e7c75>\",\"WARC-IP-Address\":\"54.221.217.175\",\"WARC-Target-URI\":\"https://socratic.org/questions/how-do-you-factor-the-expression-4x-2-16x-15\",\"WARC-Payload-Digest\":\"sha1:AZWAT6D2D2MXBHUQ4ZUHW34JUFH3ZMHH\",\"WARC-Block-Digest\":\"sha1:QJ7VG3EXIV4YP2LH2OGYCF2NZLR2O2ME\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-10/CC-MAIN-2020-10_segments_1581875144167.31_warc_CC-MAIN-20200219184416-20200219214416-00199.warc.gz\"}"}