Factor polynomial $$ 7x^{2}+13x-110 $$

$$ 7x^{2}+13x-110 = ( 7x-22 )( x+5 ) $$

Step 1: Identify constants $ a $ , $ b $ and $ c $.
$ a $ is a number in front of the $ x^2 $ term $ b $ is a number in front of the $ x $ term and $ c $ is a constant. In this case:

$$ a = 7 , b = 13 ~ \text{ and } ~ c = -110 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 7 }$ by the constant term $\color{blue}{c = -110} $.

$$ a \cdot c = -770 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -770 $ and add to $ b = 13 $.

Step 4: All pairs of numbers with a product of $ -770 $ are:

PRODUCT = -770
-1     770	1     -770
-2     385	2     -385
-5     154	5     -154
-7     110	7     -110
-10     77	10     -77
-11     70	11     -70
-14     55	14     -55
-22     35	22     -35

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 13 }$

PRODUCT = -770 and SUM = 13
-1     770	1     -770
-2     385	2     -385
-5     154	5     -154
-7     110	7     -110
-10     77	10     -77
-11     70	11     -70
-14     55	14     -55
-22     35	22     -35

Step 6: Replace middle term $ 13 x $ with $ 35x-22x $:

$$ 7x^{2}+13x-110 = 7x^{2}+35x-22x-110 $$

Step 7: Apply factoring by grouping. Factor $ 7x $ out of the first two terms and $ -22 $ out of the last two terms.

$$ 7x^{2}+35x-22x-110 = 7x\left(x+5\right) -22\left(x+5\right) = \left(7x-22\right) \left(x+5\right) $$

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