Solve left(x+7right)^2+left(-x^2-6x+7right)^2

$${ \left(x+7 \right) }^{ 2 } + { \left(- { x }^{ 2 } -6x+7 \right) }^{ 2 }$$

Evaluate

$$\left(x+7\right)^{2}\left(x^{2}-2x+2\right)$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+7\right)^{2}.

$$x^{2}+14x+49+\left(-x^{2}-6x+7\right)^{2}$$

Square -x^{2}-6x+7.

$$x^{2}+14x+49+36x^{2}+14\left(-x^{2}\right)-12x\left(-x^{2}\right)-84x+\left(-x^{2}\right)^{2}+49$$

Multiply -12 and -1 to get 12.

$$x^{2}+14x+49+36x^{2}+14\left(-x^{2}\right)+12xx^{2}-84x+\left(-x^{2}\right)^{2}+49$$

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

$$x^{2}+14x+49+36x^{2}+14\left(-x^{2}\right)+12x^{3}-84x+\left(-x^{2}\right)^{2}+49$$

Calculate -x^{2} to the power of 2 and get \left(x^{2}\right)^{2}.

$$x^{2}+14x+49+36x^{2}+14\left(-x^{2}\right)+12x^{3}-84x+\left(x^{2}\right)^{2}+49$$

Combine x^{2} and 36x^{2} to get 37x^{2}.

$$37x^{2}+14x+49+14\left(-x^{2}\right)+12x^{3}-84x+\left(x^{2}\right)^{2}+49$$

Combine 14x and -84x to get -70x.

$$37x^{2}-70x+49+14\left(-x^{2}\right)+12x^{3}+\left(x^{2}\right)^{2}+49$$

Add 49 and 49 to get 98.

$$37x^{2}-70x+98+14\left(-x^{2}\right)+12x^{3}+\left(x^{2}\right)^{2}$$

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

$$37x^{2}-70x+98+14\left(-x^{2}\right)+12x^{3}+x^{4}$$

Multiply 14 and -1 to get -14.

$$37x^{2}-70x+98-14x^{2}+12x^{3}+x^{4}$$

Combine 37x^{2} and -14x^{2} to get 23x^{2}.

$$23x^{2}-70x+98+12x^{3}+x^{4}$$

Expand

$$x^{4}+12x^{3}+23x^{2}-70x+98$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+7\right)^{2}.

$$x^{2}+14x+49+\left(-x^{2}-6x+7\right)^{2}$$

Square -x^{2}-6x+7.

$$x^{2}+14x+49+36x^{2}+14\left(-x^{2}\right)-12x\left(-x^{2}\right)-84x+\left(-x^{2}\right)^{2}+49$$

Multiply -12 and -1 to get 12.

$$x^{2}+14x+49+36x^{2}+14\left(-x^{2}\right)+12xx^{2}-84x+\left(-x^{2}\right)^{2}+49$$

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

$$x^{2}+14x+49+36x^{2}+14\left(-x^{2}\right)+12x^{3}-84x+\left(-x^{2}\right)^{2}+49$$

Calculate -x^{2} to the power of 2 and get \left(x^{2}\right)^{2}.

$$x^{2}+14x+49+36x^{2}+14\left(-x^{2}\right)+12x^{3}-84x+\left(x^{2}\right)^{2}+49$$

Combine x^{2} and 36x^{2} to get 37x^{2}.

$$37x^{2}+14x+49+14\left(-x^{2}\right)+12x^{3}-84x+\left(x^{2}\right)^{2}+49$$

Combine 14x and -84x to get -70x.

$$37x^{2}-70x+49+14\left(-x^{2}\right)+12x^{3}+\left(x^{2}\right)^{2}+49$$

Add 49 and 49 to get 98.

$$37x^{2}-70x+98+14\left(-x^{2}\right)+12x^{3}+\left(x^{2}\right)^{2}$$

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

$$37x^{2}-70x+98+14\left(-x^{2}\right)+12x^{3}+x^{4}$$

Multiply 14 and -1 to get -14.

$$37x^{2}-70x+98-14x^{2}+12x^{3}+x^{4}$$

Combine 37x^{2} and -14x^{2} to get 23x^{2}.

$$23x^{2}-70x+98+12x^{3}+x^{4}$$