Solve (x^2+1)^3-3(x^2-1)^2-5x(x-2)+10=

$$( x ^ { 2 } + 1 ) ^ { 3 } - 3 ( x ^ { 2 } - 1 ) ^ { 2 } - 5 x ( x - 2 ) + 10 =$$

Evaluate

$$x^{6}+4x^{2}+10x+8$$

Solution Steps

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}.

$$\left(x^{2}\right)^{3}+3\left(x^{2}\right)^{2}+3x^{2}+1-3\left(x^{2}-1\right)^{2}-5x\left(x-2\right)+10$$

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

$$x^{6}+3\left(x^{2}\right)^{2}+3x^{2}+1-3\left(x^{2}-1\right)^{2}-5x\left(x-2\right)+10$$

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

$$x^{6}+3x^{4}+3x^{2}+1-3\left(x^{2}-1\right)^{2}-5x\left(x-2\right)+10$$

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

$$x^{6}+3x^{4}+3x^{2}+1-3\left(\left(x^{2}\right)^{2}-2x^{2}+1\right)-5x\left(x-2\right)+10$$

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

$$x^{6}+3x^{4}+3x^{2}+1-3\left(x^{4}-2x^{2}+1\right)-5x\left(x-2\right)+10$$

Use the distributive property to multiply -3 by x^{4}-2x^{2}+1.

$$x^{6}+3x^{4}+3x^{2}+1-3x^{4}+6x^{2}-3-5x\left(x-2\right)+10$$

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

$$x^{6}+3x^{2}+1+6x^{2}-3-5x\left(x-2\right)+10$$

Combine 3x^{2} and 6x^{2} to get 9x^{2}.

$$x^{6}+9x^{2}+1-3-5x\left(x-2\right)+10$$

Subtract 3 from 1 to get -2.

$$x^{6}+9x^{2}-2-5x\left(x-2\right)+10$$

Use the distributive property to multiply -5x by x-2.

$$x^{6}+9x^{2}-2-5x^{2}+10x+10$$

Combine 9x^{2} and -5x^{2} to get 4x^{2}.

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

Add -2 and 10 to get 8.

$$x^{6}+4x^{2}+8+10x$$

Expand

$$x^{6}+4x^{2}+10x+8$$

Solution Steps

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}.

$$\left(x^{2}\right)^{3}+3\left(x^{2}\right)^{2}+3x^{2}+1-3\left(x^{2}-1\right)^{2}-5x\left(x-2\right)+10$$

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

$$x^{6}+3\left(x^{2}\right)^{2}+3x^{2}+1-3\left(x^{2}-1\right)^{2}-5x\left(x-2\right)+10$$

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

$$x^{6}+3x^{4}+3x^{2}+1-3\left(x^{2}-1\right)^{2}-5x\left(x-2\right)+10$$

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

$$x^{6}+3x^{4}+3x^{2}+1-3\left(\left(x^{2}\right)^{2}-2x^{2}+1\right)-5x\left(x-2\right)+10$$

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

$$x^{6}+3x^{4}+3x^{2}+1-3\left(x^{4}-2x^{2}+1\right)-5x\left(x-2\right)+10$$

Use the distributive property to multiply -3 by x^{4}-2x^{2}+1.

$$x^{6}+3x^{4}+3x^{2}+1-3x^{4}+6x^{2}-3-5x\left(x-2\right)+10$$

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

$$x^{6}+3x^{2}+1+6x^{2}-3-5x\left(x-2\right)+10$$

Combine 3x^{2} and 6x^{2} to get 9x^{2}.

$$x^{6}+9x^{2}+1-3-5x\left(x-2\right)+10$$

Subtract 3 from 1 to get -2.

$$x^{6}+9x^{2}-2-5x\left(x-2\right)+10$$

Use the distributive property to multiply -5x by x-2.

$$x^{6}+9x^{2}-2-5x^{2}+10x+10$$

Combine 9x^{2} and -5x^{2} to get 4x^{2}.

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

Add -2 and 10 to get 8.

$$x^{6}+4x^{2}+8+10x$$