Factor polynomial $$ -4x^{5}+8x^{4}+32x^{3} $$

$$ -4x^{5}+8x^{4}+32x^{3} = ( -4 )x^{3}( x+2 )( x-4 ) $$

Step 1 :

After factoring out $ -4x^{3} $ we have:

$$ -4x^{5}+8x^{4}+32x^{3} = -4x^{3} ( x^{2}-2x-8 ) $$

Step 2 :

Step 2: Identify constants $ \color{blue}{ b }$ and $\color{red}{ c }$. ( $ \color{blue}{ b }$ is a number in front of the $ x $ term and $ \color{red}{ c } $ is a constant). In our case:

$$ \color{blue}{ b = -2 } ~ \text{ and } ~ \color{red}{ c = -8 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -2 } $ and multiply to $ \color{red}{ -8 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = -8 }$.

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

Step 4: Find out which pair sums up to $\color{blue}{ b = -2 }$

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

Step 5: Put 2 and -4 into placeholders to get factored form.

$$ \begin{aligned} x^{2}-2x-8 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}-2x-8 & = (x + 2)(x -4) \end{aligned} $$

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