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

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

Evaluate

$$-4x^{2}+5x-12$$

Solution Steps

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-1\right)^{2}.

$$x^{2}-x-2-\left(9x^{2}-6x+1\right)+\left(2x-3\right)\left(2x+3\right)$$

To find the opposite of 9x^{2}-6x+1, find the opposite of each term.

$$x^{2}-x-2-9x^{2}+6x-1+\left(2x-3\right)\left(2x+3\right)$$

Combine x^{2} and -9x^{2} to get -8x^{2}.

$$-8x^{2}-x-2+6x-1+\left(2x-3\right)\left(2x+3\right)$$

Combine -x and 6x to get 5x.

$$-8x^{2}+5x-2-1+\left(2x-3\right)\left(2x+3\right)$$

Subtract 1 from -2 to get -3.

$$-8x^{2}+5x-3+\left(2x-3\right)\left(2x+3\right)$$

Consider \left(2x-3\right)\left(2x+3\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 3.

$$-8x^{2}+5x-3+\left(2x\right)^{2}-9$$

Expand \left(2x\right)^{2}.

$$-8x^{2}+5x-3+2^{2}x^{2}-9$$

Calculate 2 to the power of 2 and get 4.

$$-8x^{2}+5x-3+4x^{2}-9$$

Combine -8x^{2} and 4x^{2} to get -4x^{2}.

$$-4x^{2}+5x-3-9$$

Subtract 9 from -3 to get -12.

$$-4x^{2}+5x-12$$

Expand

$$-4x^{2}+5x-12$$

Solution Steps

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-1\right)^{2}.

$$x^{2}-x-2-\left(9x^{2}-6x+1\right)+\left(2x-3\right)\left(2x+3\right)$$

To find the opposite of 9x^{2}-6x+1, find the opposite of each term.

$$x^{2}-x-2-9x^{2}+6x-1+\left(2x-3\right)\left(2x+3\right)$$

Combine x^{2} and -9x^{2} to get -8x^{2}.

$$-8x^{2}-x-2+6x-1+\left(2x-3\right)\left(2x+3\right)$$

Combine -x and 6x to get 5x.

$$-8x^{2}+5x-2-1+\left(2x-3\right)\left(2x+3\right)$$

Subtract 1 from -2 to get -3.

$$-8x^{2}+5x-3+\left(2x-3\right)\left(2x+3\right)$$

Consider \left(2x-3\right)\left(2x+3\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 3.

$$-8x^{2}+5x-3+\left(2x\right)^{2}-9$$

Expand \left(2x\right)^{2}.

$$-8x^{2}+5x-3+2^{2}x^{2}-9$$

Calculate 2 to the power of 2 and get 4.

$$-8x^{2}+5x-3+4x^{2}-9$$

Combine -8x^{2} and 4x^{2} to get -4x^{2}.

$$-4x^{2}+5x-3-9$$

Subtract 9 from -3 to get -12.

$$-4x^{2}+5x-12$$