Factor polynomial $$ 2x^{2}-5x-33 $$

$$ 2x^{2}-5x-33 = ( 2x-11 )( x+3 ) $$

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:

$$ a = 2 , b = -5 ~ \text{ and } ~ c = -33 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = -33} $.

$$ a \cdot c = -66 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -66 $ and add to $ b = -5 $.

Step 4: All pairs of numbers with a product of $ -66 $ are:

PRODUCT = -66
-1     66	1     -66
-2     33	2     -33
-3     22	3     -22
-6     11	6     -11

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

PRODUCT = -66 and SUM = -5
-1     66	1     -66
-2     33	2     -33
-3     22	3     -22
-6     11	6     -11

Step 6: Replace middle term $ -5 x $ with $ 6x-11x $:

$$ 2x^{2}-5x-33 = 2x^{2}+6x-11x-33 $$

Step 7: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ -11 $ out of the last two terms.

$$ 2x^{2}+6x-11x-33 = 2x\left(x+3\right) -11\left(x+3\right) = \left(2x-11\right) \left(x+3\right) $$

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