Solve (x^2+x)^2-4(x^2+y^2)-12

$$( x ^ { 2 } + x ) ^ { 2 } - 4 ( x ^ { 2 } + y ^ { 2 } ) - 12$$

Evaluate

$$\left(x\left(x+1\right)\right)^{2}-4x^{2}-4y^{2}-12$$

Solution Steps

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

$$\left(x^{2}\right)^{2}+2x^{2}x+x^{2}-4\left(x^{2}+y^{2}\right)-12$$

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

$$x^{4}+2x^{2}x+x^{2}-4\left(x^{2}+y^{2}\right)-12$$

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

$$x^{4}+2x^{3}+x^{2}-4\left(x^{2}+y^{2}\right)-12$$

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

$$x^{4}+2x^{3}+x^{2}-4x^{2}-4y^{2}-12$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$x^{4}+2x^{3}-3x^{2}-4y^{2}-12$$

Expand

$$x^{4}+2x^{3}-3x^{2}-4y^{2}-12$$

Solution Steps

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

$$\left(x^{2}\right)^{2}+2x^{2}x+x^{2}-4\left(x^{2}+y^{2}\right)-12$$

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

$$x^{4}+2x^{2}x+x^{2}-4\left(x^{2}+y^{2}\right)-12$$

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

$$x^{4}+2x^{3}+x^{2}-4\left(x^{2}+y^{2}\right)-12$$

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

$$x^{4}+2x^{3}+x^{2}-4x^{2}-4y^{2}-12$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$x^{4}+2x^{3}-3x^{2}-4y^{2}-12$$