Simplify (x+y+1)(x-y-1)

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

Step 1

Expand $(x + y + 1) (x - y - 1)$ by multiplying each term in the first expression by each term in the second expression.

$x \cdot x + x (- y) + x \cdot - 1 + y x + y (- y) + y \cdot - 1 + 1 x + 1 (- y) + 1 \cdot - 1$

Step 2

Simplify terms.

Combine the opposite terms in $x \cdot x + x (- y) + x \cdot - 1 + y x + y (- y) + y \cdot - 1 + 1 x + 1 (- y) + 1 \cdot - 1$ .

Reorder the factors in the terms $x (- y)$ and $y x$ .

$x \cdot x - x y + x \cdot - 1 + x y + y (- y) + y \cdot - 1 + 1 x + 1 (- y) + 1 \cdot - 1$

Add $- x y$ and $x y$ .

$x \cdot x + x \cdot - 1 + 0 + y (- y) + y \cdot - 1 + 1 x + 1 (- y) + 1 \cdot - 1$

Add $x \cdot x + x \cdot - 1$ and $0$ .

$x \cdot x + x \cdot - 1 + y (- y) + y \cdot - 1 + 1 x + 1 (- y) + 1 \cdot - 1$

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 1$ to the left of $x$ .

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

Rewrite $- 1 x$ as $- x$ .

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

Rewrite using the commutative property of multiplication.

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

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

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

Multiply $y$ by $y$ .

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

Move $- 1$ to the left of $y$ .

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

Rewrite $- 1 y$ as $- y$ .

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

Multiply $x$ by $1$ .

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

Multiply $- y$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

Simplify by adding terms.

Combine the opposite terms in $x^{2} - x - y^{2} - y + x - y - 1$ .

Add $- x$ and $x$ .

$x^{2} - y^{2} - y + 0 - y - 1$

Add $x^{2} - y^{2} - y$ and $0$ .

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

Subtract $y$ from $- y$ .

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