Solve (x-2)^2-(x-2)(x+2)+(x-2)(x+1)-6=

$$( x - 2 ) ^ { 2 } - ( x - 2 ) ( x + 2 ) + ( x - 2 ) ( x + 1 ) - 6 =$$

Evaluate

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

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$x^{2}-4x+4-\left(x-2\right)\left(x+2\right)+\left(x-2\right)\left(x+1\right)-6$$

Consider \left(x-2\right)\left(x+2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 2.

$$x^{2}-4x+4-\left(x^{2}-4\right)+\left(x-2\right)\left(x+1\right)-6$$

To find the opposite of x^{2}-4, find the opposite of each term.

$$x^{2}-4x+4-x^{2}+4+\left(x-2\right)\left(x+1\right)-6$$

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

$$-4x+4+4+\left(x-2\right)\left(x+1\right)-6$$

Add 4 and 4 to get 8.

$$-4x+8+\left(x-2\right)\left(x+1\right)-6$$

Use the distributive property to multiply x-2 by x+1 and combine like terms.

$$-4x+8+x^{2}-x-2-6$$

Combine -4x and -x to get -5x.

$$-5x+8+x^{2}-2-6$$

Subtract 2 from 8 to get 6.

$$-5x+6+x^{2}-6$$

Subtract 6 from 6 to get 0.

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

Expand

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

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$x^{2}-4x+4-\left(x-2\right)\left(x+2\right)+\left(x-2\right)\left(x+1\right)-6$$

Consider \left(x-2\right)\left(x+2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 2.

$$x^{2}-4x+4-\left(x^{2}-4\right)+\left(x-2\right)\left(x+1\right)-6$$

To find the opposite of x^{2}-4, find the opposite of each term.

$$x^{2}-4x+4-x^{2}+4+\left(x-2\right)\left(x+1\right)-6$$

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

$$-4x+4+4+\left(x-2\right)\left(x+1\right)-6$$

Add 4 and 4 to get 8.

$$-4x+8+\left(x-2\right)\left(x+1\right)-6$$

Use the distributive property to multiply x-2 by x+1 and combine like terms.

$$-4x+8+x^{2}-x-2-6$$

Combine -4x and -x to get -5x.

$$-5x+8+x^{2}-2-6$$

Subtract 2 from 8 to get 6.

$$-5x+6+x^{2}-6$$

Subtract 6 from 6 to get 0.

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