Factor polynomial $$ 2x^{3}-5x^{2}-52x $$

$$ 2x^{3}-5x^{2}-52x = x( 2x-13 )( x+4 ) $$

Step 1 :

After factoring out $ x $ we have:

$$ 2x^{3}-5x^{2}-52x = x ( 2x^{2}-5x-52 ) $$

Step 2 :

Step 2: 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:

$$ a = 2 , b = -5 ~ \text{ and } ~ c = -52 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = -52} $.

$$ a \cdot c = -104 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -104 $ and add to $ b = -5 $.

Step 5: All pairs of numbers with a product of $ -104 $ are:

PRODUCT = -104
-1     104	1     -104
-2     52	2     -52
-4     26	4     -26
-8     13	8     -13

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -5 }$

PRODUCT = -104 and SUM = -5
-1     104	1     -104
-2     52	2     -52
-4     26	4     -26
-8     13	8     -13

Step 7: Replace middle term $ -5 x $ with $ 8x-13x $:

$$ 2x^{2}-5x-52 = 2x^{2}+8x-13x-52 $$

Step 8: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ -13 $ out of the last two terms.

$$ 2x^{2}+8x-13x-52 = 2x\left(x+4\right) -13\left(x+4\right) = \left(2x-13\right) \left(x+4\right) $$

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