Solve 1-(x+1)[(3x^{2}-2x)^{2}-(3x^{2}-1)(3x^2+1)+(2x-1)(4x^2+2x+1)]-(2x^2-1)^2

$$1 - ( x + 1 ) [ ( 3 x ^ { 2 } - 2 x ) ^ { 2 } - ( 3 x ^ { 2 } - 1 ) ( 3 x ^ { 2 } + 1 ) + ( 2 x - 1 ) ( 4 x ^ { 2 } + 2 x + 1 ) ] - ( 2 x ^ { 2 } - 1 ) ^ { 2 }$$

Evaluate

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Solution Steps

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

$$1-\left(x+1\right)\left(9\left(x^{2}\right)^{2}-12x^{2}x+4x^{2}-\left(3x^{2}-1\right)\left(3x^{2}+1\right)+\left(2x-1\right)\left(4x^{2}+2x+1\right)\right)-\left(2x^{2}-1\right)^{2}$$

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

$$1-\left(x+1\right)\left(9x^{4}-12x^{2}x+4x^{2}-\left(3x^{2}-1\right)\left(3x^{2}+1\right)+\left(2x-1\right)\left(4x^{2}+2x+1\right)\right)-\left(2x^{2}-1\right)^{2}$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$1-\left(x+1\right)\left(9x^{4}-12x^{3}+4x^{2}-\left(3x^{2}-1\right)\left(3x^{2}+1\right)+\left(2x-1\right)\left(4x^{2}+2x+1\right)\right)-\left(2x^{2}-1\right)^{2}$$

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

$$1-\left(x+1\right)\left(9x^{4}-12x^{3}+4x^{2}-\left(\left(3x^{2}\right)^{2}-1\right)+\left(2x-1\right)\left(4x^{2}+2x+1\right)\right)-\left(2x^{2}-1\right)^{2}$$

Expand \left(3x^{2}\right)^{2}.

$$1-\left(x+1\right)\left(9x^{4}-12x^{3}+4x^{2}-\left(3^{2}\left(x^{2}\right)^{2}-1\right)+\left(2x-1\right)\left(4x^{2}+2x+1\right)\right)-\left(2x^{2}-1\right)^{2}$$

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

$$1-\left(x+1\right)\left(9x^{4}-12x^{3}+4x^{2}-\left(3^{2}x^{4}-1\right)+\left(2x-1\right)\left(4x^{2}+2x+1\right)\right)-\left(2x^{2}-1\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

$$1-\left(x+1\right)\left(9x^{4}-12x^{3}+4x^{2}-\left(9x^{4}-1\right)+\left(2x-1\right)\left(4x^{2}+2x+1\right)\right)-\left(2x^{2}-1\right)^{2}$$

To find the opposite of 9x^{4}-1, find the opposite of each term.

$$1-\left(x+1\right)\left(9x^{4}-12x^{3}+4x^{2}-9x^{4}+1+\left(2x-1\right)\left(4x^{2}+2x+1\right)\right)-\left(2x^{2}-1\right)^{2}$$

Combine 9x^{4} and -9x^{4} to get 0.

$$1-\left(x+1\right)\left(-12x^{3}+4x^{2}+1+\left(2x-1\right)\left(4x^{2}+2x+1\right)\right)-\left(2x^{2}-1\right)^{2}$$

Use the distributive property to multiply 2x-1 by 4x^{2}+2x+1 and combine like terms.

$$1-\left(x+1\right)\left(-12x^{3}+4x^{2}+1+8x^{3}-1\right)-\left(2x^{2}-1\right)^{2}$$

Combine -12x^{3} and 8x^{3} to get -4x^{3}.

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

Subtract 1 from 1 to get 0.

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

Use the distributive property to multiply x+1 by -4x^{3}+4x^{2} and combine like terms.

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

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

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

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

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

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

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

To find the opposite of 4x^{4}-4x^{2}+1, find the opposite of each term.

$$1+4x^{4}-4x^{2}-4x^{4}+4x^{2}-1$$

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

$$1-4x^{2}+4x^{2}-1$$

Combine -4x^{2} and 4x^{2} to get 0.

$$1-1$$

Subtract 1 from 1 to get 0.

$$0$$

Factor

$$0$$