Solve -(x^{2}+9)(x-3)(3+x)-(3+x)^{2}-(4-x+x^3)^2+(x^2-1)^3-x(x+2)

$$- ( x ^ { 2 } + 9 ) ( x - 3 ) ( 3 + x ) - ( 3 + x ) ^ { 2 } - ( 4 - x + x ^ { 3 } ) ^ { 2 } + ( x ^ { 2 } - 1 ) ^ { 3 } - x ( x + 2 )$$

Evaluate

$$55-8x^{3}-2x^{4}$$

Solution Steps

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

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

Use the distributive property to multiply -x^{2}-9 by x-3.

$$\left(-x^{3}+3x^{2}-9x+27\right)\left(3+x\right)-\left(3+x\right)^{2}-\left(4-x+x^{3}\right)^{2}+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

Use the distributive property to multiply -x^{3}+3x^{2}-9x+27 by 3+x and combine like terms.

$$-x^{4}+81-\left(3+x\right)^{2}-\left(4-x+x^{3}\right)^{2}+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

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

$$-x^{4}+81-\left(9+6x+x^{2}\right)-\left(4-x+x^{3}\right)^{2}+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

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

$$-x^{4}+81-9-6x-x^{2}-\left(4-x+x^{3}\right)^{2}+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

Subtract 9 from 81 to get 72.

$$-x^{4}+72-6x-x^{2}-\left(4-x+x^{3}\right)^{2}+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

Square 4-x+x^{3}.

$$-x^{4}+72-6x-x^{2}-\left(x^{6}-2x^{4}+8x^{3}+x^{2}-8x+16\right)+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

To find the opposite of x^{6}-2x^{4}+8x^{3}+x^{2}-8x+16, find the opposite of each term.

$$-x^{4}+72-6x-x^{2}-x^{6}+2x^{4}-8x^{3}-x^{2}+8x-16+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

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

$$x^{4}+72-6x-x^{2}-x^{6}-8x^{3}-x^{2}+8x-16+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

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

$$x^{4}+72-6x-2x^{2}-x^{6}-8x^{3}+8x-16+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

Combine -6x and 8x to get 2x.

$$x^{4}+72+2x-2x^{2}-x^{6}-8x^{3}-16+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

Subtract 16 from 72 to get 56.

$$x^{4}+56+2x-2x^{2}-x^{6}-8x^{3}+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

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

$$x^{4}+56+2x-2x^{2}-x^{6}-8x^{3}+\left(x^{2}\right)^{3}-3\left(x^{2}\right)^{2}+3x^{2}-1-x\left(x+2\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$x^{4}+56+2x-2x^{2}-x^{6}-8x^{3}+x^{6}-3\left(x^{2}\right)^{2}+3x^{2}-1-x\left(x+2\right)$$

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

$$x^{4}+56+2x-2x^{2}-x^{6}-8x^{3}+x^{6}-3x^{4}+3x^{2}-1-x\left(x+2\right)$$

Combine -x^{6} and x^{6} to get 0.

$$x^{4}+56+2x-2x^{2}-8x^{3}-3x^{4}+3x^{2}-1-x\left(x+2\right)$$

Combine x^{4} and -3x^{4} to get -2x^{4}.

$$-2x^{4}+56+2x-2x^{2}-8x^{3}+3x^{2}-1-x\left(x+2\right)$$

Combine -2x^{2} and 3x^{2} to get x^{2}.

$$-2x^{4}+56+2x+x^{2}-8x^{3}-1-x\left(x+2\right)$$

Subtract 1 from 56 to get 55.

$$-2x^{4}+55+2x+x^{2}-8x^{3}-x\left(x+2\right)$$

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

$$-2x^{4}+55+2x+x^{2}-8x^{3}-\left(x^{2}+2x\right)$$

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

$$-2x^{4}+55+2x+x^{2}-8x^{3}-x^{2}-2x$$

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

$$-2x^{4}+55+2x-8x^{3}-2x$$

Combine 2x and -2x to get 0.

$$-2x^{4}+55-8x^{3}$$

Expand

$$55-8x^{3}-2x^{4}$$

Solution Steps

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

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

Use the distributive property to multiply -x^{2}-9 by x-3.

$$\left(-x^{3}+3x^{2}-9x+27\right)\left(3+x\right)-\left(3+x\right)^{2}-\left(4-x+x^{3}\right)^{2}+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

Use the distributive property to multiply -x^{3}+3x^{2}-9x+27 by 3+x and combine like terms.

$$-x^{4}+81-\left(3+x\right)^{2}-\left(4-x+x^{3}\right)^{2}+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

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

$$-x^{4}+81-\left(9+6x+x^{2}\right)-\left(4-x+x^{3}\right)^{2}+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

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

$$-x^{4}+81-9-6x-x^{2}-\left(4-x+x^{3}\right)^{2}+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

Subtract 9 from 81 to get 72.

$$-x^{4}+72-6x-x^{2}-\left(4-x+x^{3}\right)^{2}+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

Square 4-x+x^{3}.

$$-x^{4}+72-6x-x^{2}-\left(x^{6}-2x^{4}+8x^{3}+x^{2}-8x+16\right)+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

To find the opposite of x^{6}-2x^{4}+8x^{3}+x^{2}-8x+16, find the opposite of each term.

$$-x^{4}+72-6x-x^{2}-x^{6}+2x^{4}-8x^{3}-x^{2}+8x-16+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

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

$$x^{4}+72-6x-x^{2}-x^{6}-8x^{3}-x^{2}+8x-16+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

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

$$x^{4}+72-6x-2x^{2}-x^{6}-8x^{3}+8x-16+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

Combine -6x and 8x to get 2x.

$$x^{4}+72+2x-2x^{2}-x^{6}-8x^{3}-16+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

Subtract 16 from 72 to get 56.

$$x^{4}+56+2x-2x^{2}-x^{6}-8x^{3}+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

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

$$x^{4}+56+2x-2x^{2}-x^{6}-8x^{3}+\left(x^{2}\right)^{3}-3\left(x^{2}\right)^{2}+3x^{2}-1-x\left(x+2\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$x^{4}+56+2x-2x^{2}-x^{6}-8x^{3}+x^{6}-3\left(x^{2}\right)^{2}+3x^{2}-1-x\left(x+2\right)$$

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

$$x^{4}+56+2x-2x^{2}-x^{6}-8x^{3}+x^{6}-3x^{4}+3x^{2}-1-x\left(x+2\right)$$

Combine -x^{6} and x^{6} to get 0.

$$x^{4}+56+2x-2x^{2}-8x^{3}-3x^{4}+3x^{2}-1-x\left(x+2\right)$$

Combine x^{4} and -3x^{4} to get -2x^{4}.

$$-2x^{4}+56+2x-2x^{2}-8x^{3}+3x^{2}-1-x\left(x+2\right)$$

Combine -2x^{2} and 3x^{2} to get x^{2}.

$$-2x^{4}+56+2x+x^{2}-8x^{3}-1-x\left(x+2\right)$$

Subtract 1 from 56 to get 55.

$$-2x^{4}+55+2x+x^{2}-8x^{3}-x\left(x+2\right)$$

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

$$-2x^{4}+55+2x+x^{2}-8x^{3}-\left(x^{2}+2x\right)$$

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

$$-2x^{4}+55+2x+x^{2}-8x^{3}-x^{2}-2x$$

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

$$-2x^{4}+55+2x-8x^{3}-2x$$

Combine 2x and -2x to get 0.

$$-2x^{4}+55-8x^{3}$$