# Factor the following expression completely t^2 + 4tv + 4v^2.

## Question:

Factor the following expression completely:

{eq}t^2 + 4tv + 4v^2 {/eq}.

## Factoring the Following Expression:

In this case, we have to factor the expression by using the exponent rule and applying the perfect square formula.

Exponent Rule: {eq}a^mb^m=\left(ab\right)^m {/eq}

Perfect square formula: {eq}\left(a+b\right)^2=a^2+2ab+b^2 {/eq}

Given expression is {eq}t^2 + 4tv + 4v^2 {/eq}.

Let us factor the expression:

{eq}\begin{align*} t^2 + 4tv + 4v^2 &= t^2 + 2^2tv + 2^2v^2 & \left[\text{Rewrite} 4\ \text{as} 2^2 \right] \\[0.3cm] &= t^2 + 2^2tv + \left(2v\right)^2 & \left[ \text{Apply exponent rule :} a^mb^m=\left(ab\right)^m \right] \\[0.3cm] &= t^2 + 2t \cdot 2v + \left(2v\right)^2 & \left[\text{Rewrite} 2^2 t v \ \text{as} 2t \cdot 2v \right] \\[0.3cm] &= \left(t+2v\right)^2 & \left[\text{Apply Perfect Square Formula}: \quad \left(a+b\right)^2=a^2+2ab+b^2, a=t,\ b=2v \right] \\[0.3cm] \therefore t^2 + 4tv + 4v^2 &= \left(t + 2v\right)^2 \end{align*} {/eq}

Therefore, the factored expression is {eq}\boxed{ \color{blue}{ t^2 + 4tv + 4v^2 = \left(t + 2v\right)^2}} {/eq}.