Solve (1-2x)(1+2x)+(1-5x)^2-2(4x-1)^2-[-2x^2-(1-3x)^2]

$$( 1 - 2 x ) ( 1 + 2 x ) + ( 1 - 5 x ) ^ { 2 } - 2 ( 4 x - 1 ) ^ { 2 } - [ - 2 x ^ { 2 } - ( 1 - 3 x ) ^ { 2 } ]$$

Evaluate

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Solution Steps

Consider \left(1-2x\right)\left(1+2x\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.

$$1-\left(2x\right)^{2}+\left(1-5x\right)^{2}-2\left(4x-1\right)^{2}-\left(-2x^{2}-\left(1-3x\right)^{2}\right)$$

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

$$1-2^{2}x^{2}+\left(1-5x\right)^{2}-2\left(4x-1\right)^{2}-\left(-2x^{2}-\left(1-3x\right)^{2}\right)$$

Calculate 2 to the power of 2 and get 4.

$$1-4x^{2}+\left(1-5x\right)^{2}-2\left(4x-1\right)^{2}-\left(-2x^{2}-\left(1-3x\right)^{2}\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-5x\right)^{2}.

$$1-4x^{2}+1-10x+25x^{2}-2\left(4x-1\right)^{2}-\left(-2x^{2}-\left(1-3x\right)^{2}\right)$$

Add 1 and 1 to get 2.

$$2-4x^{2}-10x+25x^{2}-2\left(4x-1\right)^{2}-\left(-2x^{2}-\left(1-3x\right)^{2}\right)$$

Combine -4x^{2} and 25x^{2} to get 21x^{2}.

$$2+21x^{2}-10x-2\left(4x-1\right)^{2}-\left(-2x^{2}-\left(1-3x\right)^{2}\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4x-1\right)^{2}.

$$2+21x^{2}-10x-2\left(16x^{2}-8x+1\right)-\left(-2x^{2}-\left(1-3x\right)^{2}\right)$$

Use the distributive property to multiply -2 by 16x^{2}-8x+1.

$$2+21x^{2}-10x-32x^{2}+16x-2-\left(-2x^{2}-\left(1-3x\right)^{2}\right)$$

Combine 21x^{2} and -32x^{2} to get -11x^{2}.

$$2-11x^{2}-10x+16x-2-\left(-2x^{2}-\left(1-3x\right)^{2}\right)$$

Combine -10x and 16x to get 6x.

$$2-11x^{2}+6x-2-\left(-2x^{2}-\left(1-3x\right)^{2}\right)$$

Subtract 2 from 2 to get 0.

$$-11x^{2}+6x-\left(-2x^{2}-\left(1-3x\right)^{2}\right)$$

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

$$-11x^{2}+6x-\left(-2x^{2}-\left(1-6x+9x^{2}\right)\right)$$

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

$$-11x^{2}+6x-\left(-2x^{2}-1+6x-9x^{2}\right)$$

Combine -2x^{2} and -9x^{2} to get -11x^{2}.

$$-11x^{2}+6x-\left(-11x^{2}-1+6x\right)$$

To find the opposite of -11x^{2}-1+6x, find the opposite of each term.

$$-11x^{2}+6x+11x^{2}+1-6x$$

Combine -11x^{2} and 11x^{2} to get 0.

$$6x+1-6x$$

Combine 6x and -6x to get 0.

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Factor

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