Factor polynomial $$ -4x^{2}+4x-1 $$

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

Step 1 :

After factoring out $ -1 $ we have:

$$ -4x^{2}+4x-1 = - ~ ( 4x^{2}-4x+1 ) $$

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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