Simplify -(x+1)^2+(x+1)+1

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

Remove parentheses.

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

Simplify each term.

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

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

Expand $(x + 1) (x + 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $x$ by $1$ .

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

Multiply $x$ by $1$ .

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

Multiply $1$ by $1$ .

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

Add $x$ and $x$ .

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

Apply the distributive property.

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

Simplify.

Multiply $2$ by $- 1$ .

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

Multiply $- 1$ by $1$ .

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

Simplify by adding terms.

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

Add $- 1$ and $1$ .

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

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

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

Add $- 2 x$ and $x$ .

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