Factor polynomial $$ 5x^{2}-x-22 $$

$$ 5x^{2}-x-22 = ( 5x-11 )( x+2 ) $$

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:

$$ a = 5 , b = -1 ~ \text{ and } ~ c = -22 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 5 }$ by the constant term $\color{blue}{c = -22} $.

$$ a \cdot c = -110 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -110 $ and add to $ b = -1 $.

Step 4: All pairs of numbers with a product of $ -110 $ are:

PRODUCT = -110
-1     110	1     -110
-2     55	2     -55
-5     22	5     -22
-10     11	10     -11

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -1 }$

PRODUCT = -110 and SUM = -1
-1     110	1     -110
-2     55	2     -55
-5     22	5     -22
-10     11	10     -11

Step 6: Replace middle term $ -1 x $ with $ 10x-11x $:

$$ 5x^{2}-x-22 = 5x^{2}+10x-11x-22 $$

Step 7: Apply factoring by grouping. Factor $ 5x $ out of the first two terms and $ -11 $ out of the last two terms.

$$ 5x^{2}+10x-11x-22 = 5x\left(x+2\right) -11\left(x+2\right) = \left(5x-11\right) \left(x+2\right) $$

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