Solve [-2(2x-1)(2x+19)(x^2+x+3)]+[-(x+2)(4x+37)(2x-1)^2]

$$[ - 2 ( 2 x - 1 ) ( 2 x + 19 ) ( x ^ { 2 } + x + 3 ) ] + [ - ( x + 2 ) ( 4 x + 37 ) ( 2 x - 1 ) ^ { 2 } ]$$

Evaluate

$$40+73x-178x^{2}-244x^{3}-24x^{4}$$

Solution Steps

Use the distributive property to multiply -2 by 2x-1.

$$\left(-4x+2\right)\left(2x+19\right)\left(x^{2}+x+3\right)+\left(-\left(x+2\right)\right)\left(4x+37\right)\left(2x-1\right)^{2}$$

Use the distributive property to multiply -4x+2 by 2x+19 and combine like terms.

$$\left(-8x^{2}-72x+38\right)\left(x^{2}+x+3\right)+\left(-\left(x+2\right)\right)\left(4x+37\right)\left(2x-1\right)^{2}$$

Use the distributive property to multiply -8x^{2}-72x+38 by x^{2}+x+3 and combine like terms.

$$-8x^{4}-80x^{3}-58x^{2}-178x+114+\left(-\left(x+2\right)\right)\left(4x+37\right)\left(2x-1\right)^{2}$$

To find the opposite of x+2, find the opposite of each term.

$$-8x^{4}-80x^{3}-58x^{2}-178x+114+\left(-x-2\right)\left(4x+37\right)\left(2x-1\right)^{2}$$

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

$$-8x^{4}-80x^{3}-58x^{2}-178x+114+\left(-x-2\right)\left(4x+37\right)\left(4x^{2}-4x+1\right)$$

Use the distributive property to multiply -x-2 by 4x+37 and combine like terms.

$$-8x^{4}-80x^{3}-58x^{2}-178x+114+\left(-4x^{2}-45x-74\right)\left(4x^{2}-4x+1\right)$$

Use the distributive property to multiply -4x^{2}-45x-74 by 4x^{2}-4x+1 and combine like terms.

$$-8x^{4}-80x^{3}-58x^{2}-178x+114-16x^{4}-164x^{3}-120x^{2}+251x-74$$

Combine -8x^{4} and -16x^{4} to get -24x^{4}.

$$-24x^{4}-80x^{3}-58x^{2}-178x+114-164x^{3}-120x^{2}+251x-74$$

Combine -80x^{3} and -164x^{3} to get -244x^{3}.

$$-24x^{4}-244x^{3}-58x^{2}-178x+114-120x^{2}+251x-74$$

Combine -58x^{2} and -120x^{2} to get -178x^{2}.

$$-24x^{4}-244x^{3}-178x^{2}-178x+114+251x-74$$

Combine -178x and 251x to get 73x.

$$-24x^{4}-244x^{3}-178x^{2}+73x+114-74$$

Subtract 74 from 114 to get 40.

$$-24x^{4}-244x^{3}-178x^{2}+73x+40$$

Expand

$$40+73x-178x^{2}-244x^{3}-24x^{4}$$

Solution Steps

Use the distributive property to multiply -2 by 2x-1.

$$\left(-4x+2\right)\left(2x+19\right)\left(x^{2}+x+3\right)+\left(-\left(x+2\right)\right)\left(4x+37\right)\left(2x-1\right)^{2}$$

Use the distributive property to multiply -4x+2 by 2x+19 and combine like terms.

$$\left(-8x^{2}-72x+38\right)\left(x^{2}+x+3\right)+\left(-\left(x+2\right)\right)\left(4x+37\right)\left(2x-1\right)^{2}$$

Use the distributive property to multiply -8x^{2}-72x+38 by x^{2}+x+3 and combine like terms.

$$-8x^{4}-80x^{3}-58x^{2}-178x+114+\left(-\left(x+2\right)\right)\left(4x+37\right)\left(2x-1\right)^{2}$$

To find the opposite of x+2, find the opposite of each term.

$$-8x^{4}-80x^{3}-58x^{2}-178x+114+\left(-x-2\right)\left(4x+37\right)\left(2x-1\right)^{2}$$

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

$$-8x^{4}-80x^{3}-58x^{2}-178x+114+\left(-x-2\right)\left(4x+37\right)\left(4x^{2}-4x+1\right)$$

Use the distributive property to multiply -x-2 by 4x+37 and combine like terms.

$$-8x^{4}-80x^{3}-58x^{2}-178x+114+\left(-4x^{2}-45x-74\right)\left(4x^{2}-4x+1\right)$$

Use the distributive property to multiply -4x^{2}-45x-74 by 4x^{2}-4x+1 and combine like terms.

$$-8x^{4}-80x^{3}-58x^{2}-178x+114-16x^{4}-164x^{3}-120x^{2}+251x-74$$

Combine -8x^{4} and -16x^{4} to get -24x^{4}.

$$-24x^{4}-80x^{3}-58x^{2}-178x+114-164x^{3}-120x^{2}+251x-74$$

Combine -80x^{3} and -164x^{3} to get -244x^{3}.

$$-24x^{4}-244x^{3}-58x^{2}-178x+114-120x^{2}+251x-74$$

Combine -58x^{2} and -120x^{2} to get -178x^{2}.

$$-24x^{4}-244x^{3}-178x^{2}-178x+114+251x-74$$

Combine -178x and 251x to get 73x.

$$-24x^{4}-244x^{3}-178x^{2}+73x+114-74$$

Subtract 74 from 114 to get 40.

$$-24x^{4}-244x^{3}-178x^{2}+73x+40$$