# How do you factor completely x^2+5x+7x+35?

Nov 12, 2015

Factor by grouping to find:

${x}^{2} + 5 x + 7 x + 35 = \left(x + 7\right) \left(x + 5\right)$

#### Explanation:

${x}^{2} + 5 x + 7 x + 35 = \left({x}^{2} + 5 x\right) + \left(7 x + 35\right) = x \left(x + 5\right) + 7 \left(x + 5\right)$

$= \left(x + 7\right) \left(x + 5\right)$

With this problem, the hard work has already been done for you in the way that the middle terms are split as $5 x + 7 x$. Normally you would be faced with something like ${x}^{2} + 12 x + 35$ and have to find the split. The trick is to find two numbers which add up to the coefficient $12$ of the term in $x$ and multiply together to give the constant term $35$.

Here's another example:

${x}^{2} + 14 x + 40$

We want to find two numbers that add up to $14$ and multiply to give $40$. Having found that $10$ and $4$ work, we have:

${x}^{2} + 14 x + 40 = \left(x + 10\right) \left(x + 4\right)$

In general, we can say:

$\left(x + a\right) \left(x + b\right) = {x}^{2} + \left(a + b\right) x + a b$