Solve (x^2-2x)^2-23(x^2-2x)-112

$$( x ^ { 2 } - 2 x ) ^ { 2 } - 23 ( x ^ { 2 } - 2 x ) - 112$$

Evaluate

$$\left(x\left(x-2\right)\right)^{2}-23x\left(x-2\right)-112$$

Solution Steps

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

$$\left(x^{2}\right)^{2}-4x^{2}x+4x^{2}-23\left(x^{2}-2x\right)-112$$

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

$$x^{4}-4x^{2}x+4x^{2}-23\left(x^{2}-2x\right)-112$$

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

$$x^{4}-4x^{3}+4x^{2}-23\left(x^{2}-2x\right)-112$$

Use the distributive property to multiply -23 by x^{2}-2x.

$$x^{4}-4x^{3}+4x^{2}-23x^{2}+46x-112$$

Combine 4x^{2} and -23x^{2} to get -19x^{2}.

$$x^{4}-4x^{3}-19x^{2}+46x-112$$

Expand

$$x^{4}-4x^{3}-19x^{2}+46x-112$$

Solution Steps

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

$$\left(x^{2}\right)^{2}-4x^{2}x+4x^{2}-23\left(x^{2}-2x\right)-112$$

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

$$x^{4}-4x^{2}x+4x^{2}-23\left(x^{2}-2x\right)-112$$

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

$$x^{4}-4x^{3}+4x^{2}-23\left(x^{2}-2x\right)-112$$

Use the distributive property to multiply -23 by x^{2}-2x.

$$x^{4}-4x^{3}+4x^{2}-23x^{2}+46x-112$$

Combine 4x^{2} and -23x^{2} to get -19x^{2}.

$$x^{4}-4x^{3}-19x^{2}+46x-112$$