Step 1 :

After factoring out $ 6 $ we have:

$$ 24x^{2}+24x+6 = 6 ( 4x^{2}+4x+1 ) $$

Step 2 :

Both the first and third terms are perfect squares.

$$ 4x^2 = \left( \color{blue}{ 2x } \right)^2 ~~ \text{and} ~~ 1 = \left( \color{red}{ 1 } \right)^2 $$

The middle term ( $ 4x $ ) is two times the product of the terms that are squared.

$$ 4x = 2 \cdot \color{blue}{2x} \cdot \color{red}{1} $$

We can conclude that the polynomial $ 4x^{2}+4x+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 = 2x } $ and $ \color{red}{ B = 1 } $ so,

$$ 4x^{2}+4x+1 = ( \color{blue}{ 2x } + \color{red}{ 1 } )^2 $$

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