Solve 4+{left(2+sqrt{x}right)}^2(2-sqrt{x})+{left(2-sqrt{x}right)}^2(2+sqrt{x})=-8

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

Solve for x

$$x=7$$

Steps for Solving Linear Equation

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

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

Calculate \sqrt{x} to the power of 2 and get x.

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

Use the distributive property to multiply 4+4\sqrt{x}+x by 2-\sqrt{x} and combine like terms.

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

Calculate \sqrt{x} to the power of 2 and get x.

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

Combine -4x and 2x to get -2x.

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

Add 4 and 8 to get 12.

$$12+4\sqrt{x}-2x-x\sqrt{x}+\left(2-\sqrt{x}\right)^{2}\left(2+\sqrt{x}\right)=-8$$

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

$$12+4\sqrt{x}-2x-x\sqrt{x}+\left(4-4\sqrt{x}+\left(\sqrt{x}\right)^{2}\right)\left(2+\sqrt{x}\right)=-8$$

Calculate \sqrt{x} to the power of 2 and get x.

$$12+4\sqrt{x}-2x-x\sqrt{x}+\left(4-4\sqrt{x}+x\right)\left(2+\sqrt{x}\right)=-8$$

Use the distributive property to multiply 4-4\sqrt{x}+x by 2+\sqrt{x} and combine like terms.

$$12+4\sqrt{x}-2x-x\sqrt{x}+8-4\sqrt{x}-4\left(\sqrt{x}\right)^{2}+2x+x\sqrt{x}=-8$$

Calculate \sqrt{x} to the power of 2 and get x.

$$12+4\sqrt{x}-2x-x\sqrt{x}+8-4\sqrt{x}-4x+2x+x\sqrt{x}=-8$$

Combine -4x and 2x to get -2x.

$$12+4\sqrt{x}-2x-x\sqrt{x}+8-4\sqrt{x}-2x+x\sqrt{x}=-8$$

Add 12 and 8 to get 20.

$$20+4\sqrt{x}-2x-x\sqrt{x}-4\sqrt{x}-2x+x\sqrt{x}=-8$$

Combine 4\sqrt{x} and -4\sqrt{x} to get 0.

$$20-2x-x\sqrt{x}-2x+x\sqrt{x}=-8$$

Combine -2x and -2x to get -4x.

$$20-4x-x\sqrt{x}+x\sqrt{x}=-8$$

Combine -x\sqrt{x} and x\sqrt{x} to get 0.

$$20-4x=-8$$

Subtract 20 from both sides.

$$-4x=-8-20$$

Subtract 20 from -8 to get -28.

$$-4x=-28$$

Divide both sides by -4.

$$x=\frac{-28}{-4}$$

Divide -28 by -4 to get 7.

$$x=7$$