Solve for x (x+4)(x+1)=x-(x-2)(2-x)

$(x + 4) (x + 1) = x - (x - 2) (2 - x)$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x - (x - 2) (2 - x) = (x + 4) (x + 1)$

Step 2

Simplify $x - (x - 2) (2 - x)$ .

Simplify each term.

Apply the distributive property.

$x + (- x - - 2) (2 - x) = (x + 4) (x + 1)$

Multiply $- 1$ by $- 2$ .

$x + (- x + 2) (2 - x) = (x + 4) (x + 1)$

Expand $(- x + 2) (2 - x)$ using the FOIL Method.

Apply the distributive property.

$x - x (2 - x) + 2 (2 - x) = (x + 4) (x + 1)$

Apply the distributive property.

$x - x \cdot 2 - x (- x) + 2 (2 - x) = (x + 4) (x + 1)$

Apply the distributive property.

$x - x \cdot 2 - x (- x) + 2 \cdot 2 + 2 (- x) = (x + 4) (x + 1)$

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $- 1$ .

$x - 2 x - x (- x) + 2 \cdot 2 + 2 (- x) = (x + 4) (x + 1)$

Rewrite using the commutative property of multiplication.

$x - 2 x - 1 \cdot - 1 x \cdot x + 2 \cdot 2 + 2 (- x) = (x + 4) (x + 1)$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$x - 2 x - 1 \cdot - 1 (x \cdot x) + 2 \cdot 2 + 2 (- x) = (x + 4) (x + 1)$

Multiply $x$ by $x$ .

$x - 2 x - 1 \cdot - 1 x^{2} + 2 \cdot 2 + 2 (- x) = (x + 4) (x + 1)$

Multiply $- 1$ by $- 1$ .

$x - 2 x + 1 x^{2} + 2 \cdot 2 + 2 (- x) = (x + 4) (x + 1)$

Multiply $x^{2}$ by $1$ .

$x - 2 x + x^{2} + 2 \cdot 2 + 2 (- x) = (x + 4) (x + 1)$

Multiply $2$ by $2$ .

$x - 2 x + x^{2} + 4 + 2 (- x) = (x + 4) (x + 1)$

Multiply $- 1$ by $2$ .

$x - 2 x + x^{2} + 4 - 2 x = (x + 4) (x + 1)$

Subtract $2 x$ from $- 2 x$ .

$x - 4 x + x^{2} + 4 = (x + 4) (x + 1)$

Subtract $4 x$ from $x$ .

$- 3 x + x^{2} + 4 = (x + 4) (x + 1)$

Step 3

Simplify $(x + 4) (x + 1)$ .

Expand $(x + 4) (x + 1)$ using the FOIL Method.

Apply the distributive property.

$- 3 x + x^{2} + 4 = x (x + 1) + 4 (x + 1)$

Apply the distributive property.

$- 3 x + x^{2} + 4 = x \cdot x + x \cdot 1 + 4 (x + 1)$

Apply the distributive property.

$- 3 x + x^{2} + 4 = x \cdot x + x \cdot 1 + 4 x + 4 \cdot 1$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$- 3 x + x^{2} + 4 = x^{2} + x \cdot 1 + 4 x + 4 \cdot 1$

Multiply $x$ by $1$ .

$- 3 x + x^{2} + 4 = x^{2} + x + 4 x + 4 \cdot 1$

Multiply $4$ by $1$ .

$- 3 x + x^{2} + 4 = x^{2} + x + 4 x + 4$

Add $x$ and $4 x$ .

$- 3 x + x^{2} + 4 = x^{2} + 5 x + 4$

Step 4

Move all terms containing $x$ to the left side of the equation.

Subtract $x^{2}$ from both sides of the equation.

$- 3 x + x^{2} + 4 - x^{2} = 5 x + 4$

Subtract $5 x$ from both sides of the equation.

$- 3 x + x^{2} + 4 - x^{2} - 5 x = 4$

Combine the opposite terms in $- 3 x + x^{2} + 4 - x^{2} - 5 x$ .

Subtract $x^{2}$ from $x^{2}$ .

$- 3 x + 0 + 4 - 5 x = 4$

Add $- 3 x$ and $0$ .

$- 3 x + 4 - 5 x = 4$

Subtract $5 x$ from $- 3 x$ .

$- 8 x + 4 = 4$

Step 5

Move all terms not containing $x$ to the right side of the equation.

Subtract $4$ from both sides of the equation.

$- 8 x = 4 - 4$

Subtract $4$ from $4$ .

$- 8 x = 0$

Step 6

Divide each term in $- 8 x = 0$ by $- 8$ and simplify.

Divide each term in $- 8 x = 0$ by $- 8$ .

$\frac{- 8 x}{- 8} = \frac{0}{- 8}$

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

$\frac{- 8 x}{- 8} = \frac{0}{- 8}$

Divide $x$ by $1$ .

$x = \frac{0}{- 8}$

Simplify the right side.

Divide $0$ by $- 8$ .

$x = 0$

Solve for x (x+4)(x+1)=x-(x-2)(2-x)

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