Solve f(x)=-2(x^2+1)^2(x^2+1)(x-1)

$$f ( x ) = - 2 ( x ^ { 2 } + 1 ) ^ { 2 } ( x ^ { 2 } + 1 ) ( x - 1 )$$

Evaluate

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

Solution Steps

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

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

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}.

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

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

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

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

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

Use the distributive property to multiply -2 by x^{6}+3x^{4}+3x^{2}+1.

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

Use the distributive property to multiply -2x^{6}-6x^{4}-6x^{2}-2 by x-1.

$$-2x^{7}+2x^{6}-6x^{5}+6x^{4}-6x^{3}+6x^{2}-2x+2$$

Expand

$$2-2x+6x^{2}-6x^{3}+6x^{4}-6x^{5}+2x^{6}-2x^{7}$$

Solution Steps

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

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

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}.

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

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

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

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

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

Use the distributive property to multiply -2 by x^{6}+3x^{4}+3x^{2}+1.

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

Use the distributive property to multiply -2x^{6}-6x^{4}-6x^{2}-2 by x-1.

$$-2x^{7}+2x^{6}-6x^{5}+6x^{4}-6x^{3}+6x^{2}-2x+2$$