Solve =(x^2+3)^2+2(x-4)(x+3)+(x-4)^2

$$= ( x ^ { 2 } + 3 ) ^ { 2 } + 2 ( x - 4 ) ( x + 3 ) + ( x - 4 ) ^ { 2 }$$

Evaluate

$$x^{4}+9x^{2}-10x+1$$

Solution Steps

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

$$\left(x^{2}\right)^{2}+6x^{2}+9+2\left(x-4\right)\left(x+3\right)+\left(x-4\right)^{2}$$

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

$$x^{4}+6x^{2}+9+2\left(x-4\right)\left(x+3\right)+\left(x-4\right)^{2}$$

Use the distributive property to multiply 2 by x-4.

$$x^{4}+6x^{2}+9+\left(2x-8\right)\left(x+3\right)+\left(x-4\right)^{2}$$

Use the distributive property to multiply 2x-8 by x+3 and combine like terms.

$$x^{4}+6x^{2}+9+2x^{2}-2x-24+\left(x-4\right)^{2}$$

Combine 6x^{2} and 2x^{2} to get 8x^{2}.

$$x^{4}+8x^{2}+9-2x-24+\left(x-4\right)^{2}$$

Subtract 24 from 9 to get -15.

$$x^{4}+8x^{2}-15-2x+\left(x-4\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-4\right)^{2}.

$$x^{4}+8x^{2}-15-2x+x^{2}-8x+16$$

Combine 8x^{2} and x^{2} to get 9x^{2}.

$$x^{4}+9x^{2}-15-2x-8x+16$$

Combine -2x and -8x to get -10x.

$$x^{4}+9x^{2}-15-10x+16$$

Add -15 and 16 to get 1.

$$x^{4}+9x^{2}+1-10x$$

Expand

$$x^{4}+9x^{2}-10x+1$$

Solution Steps

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

$$\left(x^{2}\right)^{2}+6x^{2}+9+2\left(x-4\right)\left(x+3\right)+\left(x-4\right)^{2}$$

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

$$x^{4}+6x^{2}+9+2\left(x-4\right)\left(x+3\right)+\left(x-4\right)^{2}$$

Use the distributive property to multiply 2 by x-4.

$$x^{4}+6x^{2}+9+\left(2x-8\right)\left(x+3\right)+\left(x-4\right)^{2}$$

Use the distributive property to multiply 2x-8 by x+3 and combine like terms.

$$x^{4}+6x^{2}+9+2x^{2}-2x-24+\left(x-4\right)^{2}$$

Combine 6x^{2} and 2x^{2} to get 8x^{2}.

$$x^{4}+8x^{2}+9-2x-24+\left(x-4\right)^{2}$$

Subtract 24 from 9 to get -15.

$$x^{4}+8x^{2}-15-2x+\left(x-4\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-4\right)^{2}.

$$x^{4}+8x^{2}-15-2x+x^{2}-8x+16$$

Combine 8x^{2} and x^{2} to get 9x^{2}.

$$x^{4}+9x^{2}-15-2x-8x+16$$

Combine -2x and -8x to get -10x.

$$x^{4}+9x^{2}-15-10x+16$$

Add -15 and 16 to get 1.

$$x^{4}+9x^{2}+1-10x$$