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

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

Evaluate

$$8x-19$$

Solution Steps

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

$$\left(2x-4\right)\left(x+3\right)-3\left(x-1\right)^{2}+\left(x+2\right)\left(x-2\right)$$

Use the distributive property to multiply 2x-4 by x+3 and combine like terms.

$$2x^{2}+2x-12-3\left(x-1\right)^{2}+\left(x+2\right)\left(x-2\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$2x^{2}+2x-12-3\left(x^{2}-2x+1\right)+\left(x+2\right)\left(x-2\right)$$

Use the distributive property to multiply -3 by x^{2}-2x+1.

$$2x^{2}+2x-12-3x^{2}+6x-3+\left(x+2\right)\left(x-2\right)$$

Combine 2x^{2} and -3x^{2} to get -x^{2}.

$$-x^{2}+2x-12+6x-3+\left(x+2\right)\left(x-2\right)$$

Combine 2x and 6x to get 8x.

$$-x^{2}+8x-12-3+\left(x+2\right)\left(x-2\right)$$

Subtract 3 from -12 to get -15.

$$-x^{2}+8x-15+\left(x+2\right)\left(x-2\right)$$

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.

$$-x^{2}+8x-15+x^{2}-4$$

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

$$8x-15-4$$

Subtract 4 from -15 to get -19.

$$8x-19$$

Expand

$$8x-19$$

Solution Steps

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

$$\left(2x-4\right)\left(x+3\right)-3\left(x-1\right)^{2}+\left(x+2\right)\left(x-2\right)$$

Use the distributive property to multiply 2x-4 by x+3 and combine like terms.

$$2x^{2}+2x-12-3\left(x-1\right)^{2}+\left(x+2\right)\left(x-2\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$2x^{2}+2x-12-3\left(x^{2}-2x+1\right)+\left(x+2\right)\left(x-2\right)$$

Use the distributive property to multiply -3 by x^{2}-2x+1.

$$2x^{2}+2x-12-3x^{2}+6x-3+\left(x+2\right)\left(x-2\right)$$

Combine 2x^{2} and -3x^{2} to get -x^{2}.

$$-x^{2}+2x-12+6x-3+\left(x+2\right)\left(x-2\right)$$

Combine 2x and 6x to get 8x.

$$-x^{2}+8x-12-3+\left(x+2\right)\left(x-2\right)$$

Subtract 3 from -12 to get -15.

$$-x^{2}+8x-15+\left(x+2\right)\left(x-2\right)$$

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.

$$-x^{2}+8x-15+x^{2}-4$$

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

$$8x-15-4$$

Subtract 4 from -15 to get -19.

$$8x-19$$