Solve [(x-1)(x+1)-(1-2x)(1+2x)]^2-(5x^2+1)^2+(-6x)(-5)

$$[ ( x - 1 ) ( x + 1 ) - ( 1 - 2 x ) ( 1 + 2 x ) ] ^ { 2 } - ( 5 x ^ { 2 } + 1 ) ^ { 2 } + ( - 6 x ) ( - 5 )$$

Evaluate

$$3+30x-30x^{2}$$

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.

$$\left(x^{2}-1-\left(1-2x\right)\left(1+2x\right)\right)^{2}-\left(5x^{2}+1\right)^{2}-6x\left(-5\right)$$

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.

$$\left(x^{2}-1-\left(1-\left(2x\right)^{2}\right)\right)^{2}-\left(5x^{2}+1\right)^{2}-6x\left(-5\right)$$

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

$$\left(x^{2}-1-\left(1-2^{2}x^{2}\right)\right)^{2}-\left(5x^{2}+1\right)^{2}-6x\left(-5\right)$$

Calculate 2 to the power of 2 and get 4.

$$\left(x^{2}-1-\left(1-4x^{2}\right)\right)^{2}-\left(5x^{2}+1\right)^{2}-6x\left(-5\right)$$

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

$$\left(x^{2}-1-1+4x^{2}\right)^{2}-\left(5x^{2}+1\right)^{2}-6x\left(-5\right)$$

Subtract 1 from -1 to get -2.

$$\left(x^{2}-2+4x^{2}\right)^{2}-\left(5x^{2}+1\right)^{2}-6x\left(-5\right)$$

Combine x^{2} and 4x^{2} to get 5x^{2}.

$$\left(5x^{2}-2\right)^{2}-\left(5x^{2}+1\right)^{2}-6x\left(-5\right)$$

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

$$25\left(x^{2}\right)^{2}-20x^{2}+4-\left(5x^{2}+1\right)^{2}-6x\left(-5\right)$$

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

$$25x^{4}-20x^{2}+4-\left(5x^{2}+1\right)^{2}-6x\left(-5\right)$$

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

$$25x^{4}-20x^{2}+4-\left(25\left(x^{2}\right)^{2}+10x^{2}+1\right)-6x\left(-5\right)$$

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

$$25x^{4}-20x^{2}+4-\left(25x^{4}+10x^{2}+1\right)-6x\left(-5\right)$$

To find the opposite of 25x^{4}+10x^{2}+1, find the opposite of each term.

$$25x^{4}-20x^{2}+4-25x^{4}-10x^{2}-1-6x\left(-5\right)$$

Combine 25x^{4} and -25x^{4} to get 0.

$$-20x^{2}+4-10x^{2}-1-6x\left(-5\right)$$

Combine -20x^{2} and -10x^{2} to get -30x^{2}.

$$-30x^{2}+4-1-6x\left(-5\right)$$

Subtract 1 from 4 to get 3.

$$-30x^{2}+3-6x\left(-5\right)$$

Multiply -6 and -5 to get 30.

$$-30x^{2}+3+30x$$

Expand

$$3+30x-30x^{2}$$

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.

$$\left(x^{2}-1-\left(1-2x\right)\left(1+2x\right)\right)^{2}-\left(5x^{2}+1\right)^{2}-6x\left(-5\right)$$

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.

$$\left(x^{2}-1-\left(1-\left(2x\right)^{2}\right)\right)^{2}-\left(5x^{2}+1\right)^{2}-6x\left(-5\right)$$

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

$$\left(x^{2}-1-\left(1-2^{2}x^{2}\right)\right)^{2}-\left(5x^{2}+1\right)^{2}-6x\left(-5\right)$$

Calculate 2 to the power of 2 and get 4.

$$\left(x^{2}-1-\left(1-4x^{2}\right)\right)^{2}-\left(5x^{2}+1\right)^{2}-6x\left(-5\right)$$

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

$$\left(x^{2}-1-1+4x^{2}\right)^{2}-\left(5x^{2}+1\right)^{2}-6x\left(-5\right)$$

Subtract 1 from -1 to get -2.

$$\left(x^{2}-2+4x^{2}\right)^{2}-\left(5x^{2}+1\right)^{2}-6x\left(-5\right)$$

Combine x^{2} and 4x^{2} to get 5x^{2}.

$$\left(5x^{2}-2\right)^{2}-\left(5x^{2}+1\right)^{2}-6x\left(-5\right)$$

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

$$25\left(x^{2}\right)^{2}-20x^{2}+4-\left(5x^{2}+1\right)^{2}-6x\left(-5\right)$$

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

$$25x^{4}-20x^{2}+4-\left(5x^{2}+1\right)^{2}-6x\left(-5\right)$$

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

$$25x^{4}-20x^{2}+4-\left(25\left(x^{2}\right)^{2}+10x^{2}+1\right)-6x\left(-5\right)$$

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

$$25x^{4}-20x^{2}+4-\left(25x^{4}+10x^{2}+1\right)-6x\left(-5\right)$$

To find the opposite of 25x^{4}+10x^{2}+1, find the opposite of each term.

$$25x^{4}-20x^{2}+4-25x^{4}-10x^{2}-1-6x\left(-5\right)$$

Combine 25x^{4} and -25x^{4} to get 0.

$$-20x^{2}+4-10x^{2}-1-6x\left(-5\right)$$

Combine -20x^{2} and -10x^{2} to get -30x^{2}.

$$-30x^{2}+4-1-6x\left(-5\right)$$

Subtract 1 from 4 to get 3.

$$-30x^{2}+3-6x\left(-5\right)$$

Multiply -6 and -5 to get 30.

$$-30x^{2}+3+30x$$