Factor tan(x)^2+2tan(x)+1

$\tan^{2} (x) + 2 \tan (x) + 1$

Let $u = \tan (x)$ . Substitute $u$ for all occurrences of $\tan (x)$ .

$u^{2} + 2 u + 1$

Factor using the perfect square rule.

Rewrite $1$ as $1^{2}$ .

$u^{2} + 2 u + 1^{2}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 u = 2 \cdot u \cdot 1$

Rewrite the polynomial.

$u^{2} + 2 \cdot u \cdot 1 + 1^{2}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = u$ and $b = 1$ .

${(u + 1)}^{2}$

Replace all occurrences of $u$ with $\tan (x)$ .

${(\tan (x) + 1)}^{2}$