Solve (x-2)(x+2)-3(left(2x-4right)^2+5(2x+3)(1-7x)

$$(x-2)(x+2)-3( { \left(2x-4 \right) }^{ 2 } +5(2x+3)(1-7x)$$

Evaluate

$$199x^{2}+333x-97$$

Solution Steps

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

$$x^{2}-4-3\left(\left(2x-4\right)^{2}+5\left(2x+3\right)\left(1-7x\right)\right)$$

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

$$x^{2}-4-3\left(4x^{2}-16x+16+5\left(2x+3\right)\left(1-7x\right)\right)$$

Use the distributive property to multiply 5 by 2x+3.

$$x^{2}-4-3\left(4x^{2}-16x+16+\left(10x+15\right)\left(1-7x\right)\right)$$

Use the distributive property to multiply 10x+15 by 1-7x and combine like terms.

$$x^{2}-4-3\left(4x^{2}-16x+16-95x-70x^{2}+15\right)$$

Combine -16x and -95x to get -111x.

$$x^{2}-4-3\left(4x^{2}-111x+16-70x^{2}+15\right)$$

Combine 4x^{2} and -70x^{2} to get -66x^{2}.

$$x^{2}-4-3\left(-66x^{2}-111x+16+15\right)$$

Add 16 and 15 to get 31.

$$x^{2}-4-3\left(-66x^{2}-111x+31\right)$$

Use the distributive property to multiply -3 by -66x^{2}-111x+31.

$$x^{2}-4+198x^{2}+333x-93$$

Combine x^{2} and 198x^{2} to get 199x^{2}.

$$199x^{2}-4+333x-93$$

Subtract 93 from -4 to get -97.

$$199x^{2}-97+333x$$

Expand

$$199x^{2}+333x-97$$

Solution Steps

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

$$x^{2}-4-3\left(\left(2x-4\right)^{2}+5\left(2x+3\right)\left(1-7x\right)\right)$$

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

$$x^{2}-4-3\left(4x^{2}-16x+16+5\left(2x+3\right)\left(1-7x\right)\right)$$

Use the distributive property to multiply 5 by 2x+3.

$$x^{2}-4-3\left(4x^{2}-16x+16+\left(10x+15\right)\left(1-7x\right)\right)$$

Use the distributive property to multiply 10x+15 by 1-7x and combine like terms.

$$x^{2}-4-3\left(4x^{2}-16x+16-95x-70x^{2}+15\right)$$

Combine -16x and -95x to get -111x.

$$x^{2}-4-3\left(4x^{2}-111x+16-70x^{2}+15\right)$$

Combine 4x^{2} and -70x^{2} to get -66x^{2}.

$$x^{2}-4-3\left(-66x^{2}-111x+16+15\right)$$

Add 16 and 15 to get 31.

$$x^{2}-4-3\left(-66x^{2}-111x+31\right)$$

Use the distributive property to multiply -3 by -66x^{2}-111x+31.

$$x^{2}-4+198x^{2}+333x-93$$

Combine x^{2} and 198x^{2} to get 199x^{2}.

$$199x^{2}-4+333x-93$$

Subtract 93 from -4 to get -97.

$$199x^{2}-97+333x$$