Factor x^2+2x+xy+y+1

$x^{2} + 2 x + x y + y + 1$

Regroup terms.

$x y + y + x^{2} + 2 x + 1$

Factor $y$ out of $x y + y$ .

Factor $y$ out of $x y$ .

$y x + y + x^{2} + 2 x + 1$

Raise $y$ to the power of $1$ .

$y x + y^{1} + x^{2} + 2 x + 1$

Factor $y$ out of $y^{1}$ .

$y x + y \cdot 1 + x^{2} + 2 x + 1$

Factor $y$ out of $y x + y \cdot 1$ .

$y (x + 1) + x^{2} + 2 x + 1$

Factor using the perfect square rule.

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

$y (x + 1) + x^{2} + 2 x + 1^{2}$

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

$2 x = 2 \cdot x \cdot 1$

Rewrite the polynomial.

$y (x + 1) + x^{2} + 2 \cdot x \cdot 1 + 1^{2}$

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

$y (x + 1) + {(x + 1)}^{2}$

Factor $x + 1$ out of $y (x + 1) + {(x + 1)}^{2}$ .

Factor $x + 1$ out of $y (x + 1)$ .

$(x + 1) y + {(x + 1)}^{2}$

Factor $x + 1$ out of ${(x + 1)}^{2}$ .

$(x + 1) y + (x + 1) (x + 1)$

Factor $x + 1$ out of $(x + 1) y + (x + 1) (x + 1)$ .

$(x + 1) (y + x + 1)$