Solve left(+--(x^2+9right)left(x-3right)left(3+xright)-left(3+xright)^2-left(4-x+x^3right)^2+{left({x}^2-1right)}^3-xleft(x+2right)

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

Evaluate

$$-8x^{3}-107$$

Solution Steps

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

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

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 -90.

$$x^{4}-90-6x-x^{2}-\left(4-x+x^{3}\right)^{2}+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

Square 4-x+x^{3}.

$$x^{4}-90-6x-x^{2}-\left(x^{6}-2x^{4}+8x^{3}+x^{2}-8x+16\right)+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

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

$$x^{4}-90-6x-x^{2}-x^{6}+2x^{4}-8x^{3}-x^{2}+8x-16+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

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

$$3x^{4}-90-6x-x^{2}-x^{6}-8x^{3}-x^{2}+8x-16+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

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

$$3x^{4}-90-6x-2x^{2}-x^{6}-8x^{3}+8x-16+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

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

$$3x^{4}-90+2x-2x^{2}-x^{6}-8x^{3}-16+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

Subtract 16 from -90 to get -106.

$$3x^{4}-106+2x-2x^{2}-x^{6}-8x^{3}+\left(x^{2}-1\right)^{3}-x\left(x+2\right)$$

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}.

$$3x^{4}-106+2x-2x^{2}-x^{6}-8x^{3}+\left(x^{2}\right)^{3}-3\left(x^{2}\right)^{2}+3x^{2}-1-x\left(x+2\right)$$

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

$$3x^{4}-106+2x-2x^{2}-x^{6}-8x^{3}+x^{6}-3\left(x^{2}\right)^{2}+3x^{2}-1-x\left(x+2\right)$$

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

$$3x^{4}-106+2x-2x^{2}-x^{6}-8x^{3}+x^{6}-3x^{4}+3x^{2}-1-x\left(x+2\right)$$

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

$$3x^{4}-106+2x-2x^{2}-8x^{3}-3x^{4}+3x^{2}-1-x\left(x+2\right)$$

Combine 3x^{4} and -3x^{4} to get 0.

$$-106+2x-2x^{2}-8x^{3}+3x^{2}-1-x\left(x+2\right)$$

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

$$-106+2x+x^{2}-8x^{3}-1-x\left(x+2\right)$$

Subtract 1 from -106 to get -107.

$$-107+2x+x^{2}-8x^{3}-x\left(x+2\right)$$

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

$$-107+2x+x^{2}-8x^{3}-\left(x^{2}+2x\right)$$

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

$$-107+2x+x^{2}-8x^{3}-x^{2}-2x$$

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

$$-107+2x-8x^{3}-2x$$

Combine 2x and -2x to get 0.

$$-107-8x^{3}$$

Differentiate w.r.t. x

$$-24x^{2}$$

Steps Using Derivative Rule for Sum

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

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

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

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

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

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

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

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

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

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

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

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

Subtract 9 from -81 to get -90.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{4}-90-6x-x^{2}-\left(4-x+x^{3}\right)^{2}+\left(x^{2}-1\right)^{3}-x\left(x+2\right))$$

Square 4-x+x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{4}-90-6x-x^{2}-\left(x^{6}-2x^{4}+8x^{3}+x^{2}-8x+16\right)+\left(x^{2}-1\right)^{3}-x\left(x+2\right))$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{4}-90-6x-x^{2}-x^{6}+2x^{4}-8x^{3}-x^{2}+8x-16+\left(x^{2}-1\right)^{3}-x\left(x+2\right))$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{4}-90-6x-x^{2}-x^{6}-8x^{3}-x^{2}+8x-16+\left(x^{2}-1\right)^{3}-x\left(x+2\right))$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{4}-90-6x-2x^{2}-x^{6}-8x^{3}+8x-16+\left(x^{2}-1\right)^{3}-x\left(x+2\right))$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{4}-90+2x-2x^{2}-x^{6}-8x^{3}-16+\left(x^{2}-1\right)^{3}-x\left(x+2\right))$$

Subtract 16 from -90 to get -106.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{4}-106+2x-2x^{2}-x^{6}-8x^{3}+\left(x^{2}-1\right)^{3}-x\left(x+2\right))$$

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}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{4}-106+2x-2x^{2}-x^{6}-8x^{3}+\left(x^{2}\right)^{3}-3\left(x^{2}\right)^{2}+3x^{2}-1-x\left(x+2\right))$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{4}-106+2x-2x^{2}-x^{6}-8x^{3}+x^{6}-3\left(x^{2}\right)^{2}+3x^{2}-1-x\left(x+2\right))$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{4}-106+2x-2x^{2}-x^{6}-8x^{3}+x^{6}-3x^{4}+3x^{2}-1-x\left(x+2\right))$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{4}-106+2x-2x^{2}-8x^{3}-3x^{4}+3x^{2}-1-x\left(x+2\right))$$

Combine 3x^{4} and -3x^{4} to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-106+2x-2x^{2}-8x^{3}+3x^{2}-1-x\left(x+2\right))$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(-106+2x+x^{2}-8x^{3}-1-x\left(x+2\right))$$

Subtract 1 from -106 to get -107.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-107+2x+x^{2}-8x^{3}-x\left(x+2\right))$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(-107+2x+x^{2}-8x^{3}-\left(x^{2}+2x\right))$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(-107+2x+x^{2}-8x^{3}-x^{2}-2x)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(-107+2x-8x^{3}-2x)$$

Combine 2x and -2x to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-107-8x^{3})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$3\left(-8\right)x^{3-1}$$

Multiply 3 times -8.

$$-24x^{3-1}$$

Subtract 1 from 3.

$$-24x^{2}$$