Solve (x^{2}+2)^3(2x)-2x^2(3)(x^2+2)^2(2x)

$$( x ^ { 2 } + 2 ) ^ { 3 } ( 2 x ) - 2 x ^ { 2 } ( 3 ) ( x ^ { 2 } + 2 ) ^ { 2 } ( 2 x )$$

Evaluate

$$2x\left(2-5x^{2}\right)\left(x^{2}+2\right)^{2}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

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

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

$$\left(\left(x^{2}\right)^{3}+6\left(x^{2}\right)^{2}+12x^{2}+8\right)\times 2x-2x^{3}\times 3\left(x^{2}+2\right)^{2}\times 2$$

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

$$\left(x^{6}+6\left(x^{2}\right)^{2}+12x^{2}+8\right)\times 2x-2x^{3}\times 3\left(x^{2}+2\right)^{2}\times 2$$

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

$$\left(x^{6}+6x^{4}+12x^{2}+8\right)\times 2x-2x^{3}\times 3\left(x^{2}+2\right)^{2}\times 2$$

Use the distributive property to multiply x^{6}+6x^{4}+12x^{2}+8 by 2.

$$\left(2x^{6}+12x^{4}+24x^{2}+16\right)x-2x^{3}\times 3\left(x^{2}+2\right)^{2}\times 2$$

Use the distributive property to multiply 2x^{6}+12x^{4}+24x^{2}+16 by x.

$$2x^{7}+12x^{5}+24x^{3}+16x-2x^{3}\times 3\left(x^{2}+2\right)^{2}\times 2$$

Multiply 2 and 3 to get 6.

$$2x^{7}+12x^{5}+24x^{3}+16x-6x^{3}\left(x^{2}+2\right)^{2}\times 2$$

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

$$2x^{7}+12x^{5}+24x^{3}+16x-6x^{3}\left(\left(x^{2}\right)^{2}+4x^{2}+4\right)\times 2$$

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

$$2x^{7}+12x^{5}+24x^{3}+16x-6x^{3}\left(x^{4}+4x^{2}+4\right)\times 2$$

Multiply 6 and 2 to get 12.

$$2x^{7}+12x^{5}+24x^{3}+16x-12x^{3}\left(x^{4}+4x^{2}+4\right)$$

Use the distributive property to multiply -12x^{3} by x^{4}+4x^{2}+4.

$$2x^{7}+12x^{5}+24x^{3}+16x-12x^{7}-48x^{5}-48x^{3}$$

Combine 2x^{7} and -12x^{7} to get -10x^{7}.

$$-10x^{7}+12x^{5}+24x^{3}+16x-48x^{5}-48x^{3}$$

Combine 12x^{5} and -48x^{5} to get -36x^{5}.

$$-10x^{7}-36x^{5}+24x^{3}+16x-48x^{3}$$

Combine 24x^{3} and -48x^{3} to get -24x^{3}.

$$-10x^{7}-36x^{5}-24x^{3}+16x$$

Expand

$$16x-24x^{3}-36x^{5}-10x^{7}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

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

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

$$\left(\left(x^{2}\right)^{3}+6\left(x^{2}\right)^{2}+12x^{2}+8\right)\times 2x-2x^{3}\times 3\left(x^{2}+2\right)^{2}\times 2$$

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

$$\left(x^{6}+6\left(x^{2}\right)^{2}+12x^{2}+8\right)\times 2x-2x^{3}\times 3\left(x^{2}+2\right)^{2}\times 2$$

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

$$\left(x^{6}+6x^{4}+12x^{2}+8\right)\times 2x-2x^{3}\times 3\left(x^{2}+2\right)^{2}\times 2$$

Use the distributive property to multiply x^{6}+6x^{4}+12x^{2}+8 by 2.

$$\left(2x^{6}+12x^{4}+24x^{2}+16\right)x-2x^{3}\times 3\left(x^{2}+2\right)^{2}\times 2$$

Use the distributive property to multiply 2x^{6}+12x^{4}+24x^{2}+16 by x.

$$2x^{7}+12x^{5}+24x^{3}+16x-2x^{3}\times 3\left(x^{2}+2\right)^{2}\times 2$$

Multiply 2 and 3 to get 6.

$$2x^{7}+12x^{5}+24x^{3}+16x-6x^{3}\left(x^{2}+2\right)^{2}\times 2$$

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

$$2x^{7}+12x^{5}+24x^{3}+16x-6x^{3}\left(\left(x^{2}\right)^{2}+4x^{2}+4\right)\times 2$$

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

$$2x^{7}+12x^{5}+24x^{3}+16x-6x^{3}\left(x^{4}+4x^{2}+4\right)\times 2$$

Multiply 6 and 2 to get 12.

$$2x^{7}+12x^{5}+24x^{3}+16x-12x^{3}\left(x^{4}+4x^{2}+4\right)$$

Use the distributive property to multiply -12x^{3} by x^{4}+4x^{2}+4.

$$2x^{7}+12x^{5}+24x^{3}+16x-12x^{7}-48x^{5}-48x^{3}$$

Combine 2x^{7} and -12x^{7} to get -10x^{7}.

$$-10x^{7}+12x^{5}+24x^{3}+16x-48x^{5}-48x^{3}$$

Combine 12x^{5} and -48x^{5} to get -36x^{5}.

$$-10x^{7}-36x^{5}+24x^{3}+16x-48x^{3}$$

Combine 24x^{3} and -48x^{3} to get -24x^{3}.

$$-10x^{7}-36x^{5}-24x^{3}+16x$$