Factor polynomial $$ x^{5}+6x^{4}+5x^{3} $$

$$ x^{5}+6x^{4}+5x^{3} = x^{3}( x+1 )( x+5 ) $$

Step 1 :

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

$$ x^{5}+6x^{4}+5x^{3} = x^{3} ( x^{2}+6x+5 ) $$

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:

$$ \color{blue}{ b = 6 } ~ \text{ and } ~ \color{red}{ c = 5 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ 6 } $ and multiply to $ \color{red}{ 5 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = 5 }$.

PRODUCT = 5
1     5	-1     -5

Step 4: Find out which pair sums up to $\color{blue}{ b = 6 }$

PRODUCT = 5 and SUM = 6
1     5	-1     -5

Step 5: Put 1 and 5 into placeholders to get factored form.

$$ \begin{aligned} x^{2}+6x+5 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}+6x+5 & = (x + 1)(x + 5) \end{aligned} $$

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