Solve (x-1)(x-2)-(x+1)(x-3)-2(x-2)(x+2)

$$(x-1)(x-2)-(x+1)(x-3)-2(x-2)(x+2)$$

Evaluate

$$13-x-2x^{2}$$

Solution Steps

Apply the distributive property by multiplying each term of x-1 by each term of x-2.

$$x^{2}-2x-x+2-\left(x+1\right)\left(x-3\right)-2\left(x-2\right)\left(x+2\right)$$

Combine -2x and -x to get -3x.

$$x^{2}-3x+2-\left(x+1\right)\left(x-3\right)-2\left(x-2\right)\left(x+2\right)$$

Apply the distributive property by multiplying each term of x+1 by each term of x-3.

$$x^{2}-3x+2-\left(x^{2}-3x+x-3\right)-2\left(x-2\right)\left(x+2\right)$$

Combine -3x and x to get -2x.

$$x^{2}-3x+2-\left(x^{2}-2x-3\right)-2\left(x-2\right)\left(x+2\right)$$

To find the opposite of x^{2}-2x-3, find the opposite of each term.

$$x^{2}-3x+2-x^{2}-\left(-2x\right)-\left(-3\right)-2\left(x-2\right)\left(x+2\right)$$

The opposite of -2x is 2x.

$$x^{2}-3x+2-x^{2}+2x-\left(-3\right)-2\left(x-2\right)\left(x+2\right)$$

The opposite of -3 is 3.

$$x^{2}-3x+2-x^{2}+2x+3-2\left(x-2\right)\left(x+2\right)$$

Combine x^{2} and -x^{2} to get 0.

$$-3x+2+2x+3-2\left(x-2\right)\left(x+2\right)$$

Combine -3x and 2x to get -x.

$$-x+2+3-2\left(x-2\right)\left(x+2\right)$$

Add 2 and 3 to get 5.

$$-x+5-2\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply -2 by x-2.

$$-x+5+\left(-2x+4\right)\left(x+2\right)$$

Apply the distributive property by multiplying each term of -2x+4 by each term of x+2.

$$-x+5-2x^{2}-4x+4x+8$$

Combine -4x and 4x to get 0.

$$-x+5-2x^{2}+8$$

Add 5 and 8 to get 13.

$$-x+13-2x^{2}$$

Expand

$$13-x-2x^{2}$$

Solution Steps

Apply the distributive property by multiplying each term of x-1 by each term of x-2.

$$x^{2}-2x-x+2-\left(x+1\right)\left(x-3\right)-2\left(x-2\right)\left(x+2\right)$$

Combine -2x and -x to get -3x.

$$x^{2}-3x+2-\left(x+1\right)\left(x-3\right)-2\left(x-2\right)\left(x+2\right)$$

Apply the distributive property by multiplying each term of x+1 by each term of x-3.

$$x^{2}-3x+2-\left(x^{2}-3x+x-3\right)-2\left(x-2\right)\left(x+2\right)$$

Combine -3x and x to get -2x.

$$x^{2}-3x+2-\left(x^{2}-2x-3\right)-2\left(x-2\right)\left(x+2\right)$$

To find the opposite of x^{2}-2x-3, find the opposite of each term.

$$x^{2}-3x+2-x^{2}-\left(-2x\right)-\left(-3\right)-2\left(x-2\right)\left(x+2\right)$$

The opposite of -2x is 2x.

$$x^{2}-3x+2-x^{2}+2x-\left(-3\right)-2\left(x-2\right)\left(x+2\right)$$

The opposite of -3 is 3.

$$x^{2}-3x+2-x^{2}+2x+3-2\left(x-2\right)\left(x+2\right)$$

Combine x^{2} and -x^{2} to get 0.

$$-3x+2+2x+3-2\left(x-2\right)\left(x+2\right)$$

Combine -3x and 2x to get -x.

$$-x+2+3-2\left(x-2\right)\left(x+2\right)$$

Add 2 and 3 to get 5.

$$-x+5-2\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply -2 by x-2.

$$-x+5+\left(-2x+4\right)\left(x+2\right)$$

Apply the distributive property by multiplying each term of -2x+4 by each term of x+2.

$$-x+5-2x^{2}-4x+4x+8$$

Combine -4x and 4x to get 0.

$$-x+5-2x^{2}+8$$

Add 5 and 8 to get 13.

$$-x+13-2x^{2}$$