Factor polynomial $$ x^{3}+2x^{2}+x $$

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

Step 1 :

After factoring out $ x $ we have:

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

Step 2 :

Both the first and third terms are perfect squares.

$$ x^2 = \left( \color{blue}{ x } \right)^2 ~~ \text{and} ~~ 1 = \left( \color{red}{ 1 } \right)^2 $$

The middle term ( $ 2x $ ) is two times the product of the terms that are squared.

$$ 2x = 2 \cdot \color{blue}{x} \cdot \color{red}{1} $$

We can conclude that the polynomial $ x^{2}+2x+1 $ is a perfect square trinomial, so we will use the formula below.

$$ A^2 + 2AB + B^2 = (A + B)^2 $$

In this example we have $ \color{blue}{ A = x } $ and $ \color{red}{ B = 1 } $ so,

$$ x^{2}+2x+1 = ( \color{blue}{ x } + \color{red}{ 1 } )^2 $$

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