Solve [(x+1)(x-1)]^2-(2+x^2)^2+3/2(2x-3)(2x+3)

$$[ ( x + 1 ) ( x - 1 ) ] ^ { 2 } - ( 2 + x ^ { 2 } ) ^ { 2 } + \frac { 3 } { 2 } ( 2 x - 3 ) ( 2 x + 3 )$$

Evaluate

$$-\frac{33}{2}=-16.5$$

Solution Steps

Consider \left(x+1\right)\left(x-1\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 1.

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

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

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

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

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

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

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

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

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

Subtract 4 from 1 to get -3.

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

Combine -2x^{2} and -4x^{2} to get -6x^{2}.

$$x^{4}-6x^{2}-3-x^{4}+\frac{3}{2}\left(2x-3\right)\left(2x+3\right)$$

Combine x^{4} and -x^{4} to get 0.

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

Use the distributive property to multiply \frac{3}{2} by 2x-3.

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

Use the distributive property to multiply 3x-\frac{9}{2} by 2x+3 and combine like terms.

$$-6x^{2}-3+6x^{2}-\frac{27}{2}$$

Combine -6x^{2} and 6x^{2} to get 0.

$$-3-\frac{27}{2}$$

Subtract \frac{27}{2} from -3 to get -\frac{33}{2}.

$$-\frac{33}{2}$$

Factor

$$-\frac{33}{2} = -16\frac{1}{2} = -16.5$$

Solution Steps

Factor out \frac{1}{2}.

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

Simplify.

$$-\frac{33}{2}$$