Factor polynomial $$ 27x^{2}+126x+147 $$

$$ 27x^{2}+126x+147 = 3( 3x+7 )^{ 2 } $$

Step 1 :

After factoring out $ 3 $ we have:

$$ 27x^{2}+126x+147 = 3 ( 9x^{2}+42x+49 ) $$

Step 2 :

Both the first and third terms are perfect squares.

$$ 9x^2 = \left( \color{blue}{ 3x } \right)^2 ~~ \text{and} ~~ 49 = \left( \color{red}{ 7 } \right)^2 $$

The middle term ( $ 42x $ ) is two times the product of the terms that are squared.

$$ 42x = 2 \cdot \color{blue}{3x} \cdot \color{red}{7} $$

We can conclude that the polynomial $ 9x^{2}+42x+49 $ 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 = 3x } $ and $ \color{red}{ B = 7 } $ so,

$$ 9x^{2}+42x+49 = ( \color{blue}{ 3x } + \color{red}{ 7 } )^2 $$

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