Factor polynomial $$ 18a^{2}-48a+32 $$

$$ 18a^{2}-48a+32 = 2( 3a-4 )^{ 2 } $$

Step 1 :

After factoring out $ 2 $ we have:

$$ 18a^{2}-48a+32 = 2 ( 9a^{2}-24a+16 ) $$

Step 2 :

Both the first and third terms are perfect squares.

$$ 9x^2 = \left( \color{blue}{ 3a } \right)^2 ~~ \text{and} ~~ 16 = \left( \color{red}{ 4 } \right)^2 $$

The middle term ( $ -24x $ ) is two times the product of the terms that are squared.

$$ -24x = - 2 \cdot \color{blue}{3a} \cdot \color{red}{4} $$

We can conclude that the polynomial $ 9a^{2}-24a+16 $ 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 = 3a } $ and $ \color{red}{ B = 4 } $ so,

$$ 9a^{2}-24a+16 = ( \color{blue}{ 3a } - \color{red}{ 4 } )^2 $$

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