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

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

Evaluate

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

Solution Steps

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

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

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

$$x^{6}-9\left(x^{2}\right)^{2}+27x^{2}-27-\left(x-2\right)\left(x^{2}+4\right)\left(x+2\right)$$

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

$$x^{6}-9x^{4}+27x^{2}-27-\left(x-2\right)\left(x^{2}+4\right)\left(x+2\right)$$

Use the distributive property to multiply x-2 by x^{2}+4.

$$x^{6}-9x^{4}+27x^{2}-27-\left(x^{3}+4x-2x^{2}-8\right)\left(x+2\right)$$

Use the distributive property to multiply x^{3}+4x-2x^{2}-8 by x+2 and combine like terms.

$$x^{6}-9x^{4}+27x^{2}-27-\left(x^{4}-16\right)$$

To find the opposite of x^{4}-16, find the opposite of each term.

$$x^{6}-9x^{4}+27x^{2}-27-x^{4}+16$$

Combine -9x^{4} and -x^{4} to get -10x^{4}.

$$x^{6}-10x^{4}+27x^{2}-27+16$$

Add -27 and 16 to get -11.

$$x^{6}-10x^{4}+27x^{2}-11$$

Expand

$$x^{6}-10x^{4}+27x^{2}-11$$

Solution Steps

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

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

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

$$x^{6}-9\left(x^{2}\right)^{2}+27x^{2}-27-\left(x-2\right)\left(x^{2}+4\right)\left(x+2\right)$$

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

$$x^{6}-9x^{4}+27x^{2}-27-\left(x-2\right)\left(x^{2}+4\right)\left(x+2\right)$$

Use the distributive property to multiply x-2 by x^{2}+4.

$$x^{6}-9x^{4}+27x^{2}-27-\left(x^{3}+4x-2x^{2}-8\right)\left(x+2\right)$$

Use the distributive property to multiply x^{3}+4x-2x^{2}-8 by x+2 and combine like terms.

$$x^{6}-9x^{4}+27x^{2}-27-\left(x^{4}-16\right)$$

To find the opposite of x^{4}-16, find the opposite of each term.

$$x^{6}-9x^{4}+27x^{2}-27-x^{4}+16$$

Combine -9x^{4} and -x^{4} to get -10x^{4}.

$$x^{6}-10x^{4}+27x^{2}-27+16$$

Add -27 and 16 to get -11.

$$x^{6}-10x^{4}+27x^{2}-11$$