Factor polynomial $$ 64a^{4}-224a^{2}+196 $$

$$ 64a^{4}-224a^{2}+196 = 4( 4a^{2}-7 )^{ 2 } $$

Step 1 :

After factoring out $ 4 $ we have:

$$ 64a^{4}-224a^{2}+196 = 4 ( 16a^{4}-56a^{2}+49 ) $$

Step 2 :

Both the first and third terms are perfect squares.

$$ 16x^4 = \left( \color{blue}{ 4a^{2} } \right)^2 ~~ \text{and} ~~ 49 = \left( \color{red}{ 7 } \right)^2 $$

The middle term ( $ -56x^2 $ ) is two times the product of the terms that are squared.

$$ -56x^2 = - 2 \cdot \color{blue}{4a^{2}} \cdot \color{red}{7} $$

We can conclude that the polynomial $ 16a^{4}-56a^{2}+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 = 4a^{2} } $ and $ \color{red}{ B = 7 } $ so,

$$ 16a^{4}-56a^{2}+49 = ( \color{blue}{ 4a^{2} } - \color{red}{ 7 } )^2 $$

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