Factor polynomial $$ 4x^{5}-7x^{4}-2x^{3} $$

$$ 4x^{5}-7x^{4}-2x^{3} = x^{3}( x-2 )( 4x+1 ) $$

Step 1 :

After factoring out $ x^{3} $ we have:

$$ 4x^{5}-7x^{4}-2x^{3} = x^{3} ( 4x^{2}-7x-2 ) $$

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:

$$ a = 4 , b = -7 ~ \text{ and } ~ c = -2 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 4 }$ by the constant term $\color{blue}{c = -2} $.

$$ a \cdot c = -8 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -8 $ and add to $ b = -7 $.

Step 5: All pairs of numbers with a product of $ -8 $ are:

PRODUCT = -8
-1     8	1     -8
-2     4	2     -4

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -7 }$

PRODUCT = -8 and SUM = -7
-1     8	1     -8
-2     4	2     -4

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

$$ 4x^{2}-7x-2 = 4x^{2}+x-8x-2 $$

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

$$ 4x^{2}+x-8x-2 = x\left(4x+1\right) -2\left(4x+1\right) = \left(x-2\right) \left(4x+1\right) $$

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