Factor polynomial $$ x^{4}+8x^{3}+7x^{2} $$

$$ x^{4}+8x^{3}+7x^{2} = x^{2}( x+1 )( x+7 ) $$

Step 1 :

After factoring out $ x^{2} $ we have:

$$ x^{4}+8x^{3}+7x^{2} = x^{2} ( x^{2}+8x+7 ) $$

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:

$$ \color{blue}{ b = 8 } ~ \text{ and } ~ \color{red}{ c = 7 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ 8 } $ and multiply to $ \color{red}{ 7 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = 7 }$.

PRODUCT = 7
1     7	-1     -7

Step 4: Find out which pair sums up to $\color{blue}{ b = 8 }$

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

Step 5: Put 1 and 7 into placeholders to get factored form.

$$ \begin{aligned} x^{2}+8x+7 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}+8x+7 & = (x + 1)(x + 7) \end{aligned} $$

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