Solve (x^2+2xsqrt{x^2+24}+x^2+24)*(x^2-2xsqrt{{x}^2+24}+{x}^2+24)

$$( { x }^{ 2 } +2x \sqrt{ { x }^{ 2 } +24 } + { x }^{ 2 } +24) \times ( { x }^{ 2 } -2x \sqrt{ { x }^{ 2 } +24 } + { x }^{ 2 } +24)$$

Evaluate

$$576$$

Solution Steps

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

$$\left(2x^{2}+2x\sqrt{x^{2}+24}+24\right)\left(x^{2}-2x\sqrt{x^{2}+24}+x^{2}+24\right)$$

Use the distributive property to multiply 2x^{2}+2x\sqrt{x^{2}+24}+24 by x^{2}-2x\sqrt{x^{2}+24}+x^{2}+24 and combine like terms.

$$2x^{2}\left(x^{2}-2x\sqrt{x^{2}+24}\right)+2x^{4}+72x^{2}+2x\sqrt{x^{2}+24}\left(x^{2}-2x\sqrt{x^{2}+24}\right)+2\sqrt{x^{2}+24}x^{3}+48\sqrt{x^{2}+24}x+24\left(x^{2}-2x\sqrt{x^{2}+24}\right)+576$$

Use the distributive property to multiply 2x^{2} by x^{2}-2x\sqrt{x^{2}+24}.

$$2x^{4}-4\sqrt{x^{2}+24}x^{3}+2x^{4}+72x^{2}+2x\sqrt{x^{2}+24}\left(x^{2}-2x\sqrt{x^{2}+24}\right)+2\sqrt{x^{2}+24}x^{3}+48\sqrt{x^{2}+24}x+24\left(x^{2}-2x\sqrt{x^{2}+24}\right)+576$$

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

$$4x^{4}-4\sqrt{x^{2}+24}x^{3}+72x^{2}+2x\sqrt{x^{2}+24}\left(x^{2}-2x\sqrt{x^{2}+24}\right)+2\sqrt{x^{2}+24}x^{3}+48\sqrt{x^{2}+24}x+24\left(x^{2}-2x\sqrt{x^{2}+24}\right)+576$$

Use the distributive property to multiply 2x\sqrt{x^{2}+24} by x^{2}-2x\sqrt{x^{2}+24}.

$$4x^{4}-4\sqrt{x^{2}+24}x^{3}+72x^{2}+2\sqrt{x^{2}+24}x^{3}-4\left(\sqrt{x^{2}+24}\right)^{2}x^{2}+2\sqrt{x^{2}+24}x^{3}+48\sqrt{x^{2}+24}x+24\left(x^{2}-2x\sqrt{x^{2}+24}\right)+576$$

Calculate \sqrt{x^{2}+24} to the power of 2 and get x^{2}+24.

$$4x^{4}-4\sqrt{x^{2}+24}x^{3}+72x^{2}+2\sqrt{x^{2}+24}x^{3}-4\left(x^{2}+24\right)x^{2}+2\sqrt{x^{2}+24}x^{3}+48\sqrt{x^{2}+24}x+24\left(x^{2}-2x\sqrt{x^{2}+24}\right)+576$$

Use the distributive property to multiply -4 by x^{2}+24.

$$4x^{4}-4\sqrt{x^{2}+24}x^{3}+72x^{2}+2\sqrt{x^{2}+24}x^{3}+\left(-4x^{2}-96\right)x^{2}+2\sqrt{x^{2}+24}x^{3}+48\sqrt{x^{2}+24}x+24\left(x^{2}-2x\sqrt{x^{2}+24}\right)+576$$

Use the distributive property to multiply -4x^{2}-96 by x^{2}.

$$4x^{4}-4\sqrt{x^{2}+24}x^{3}+72x^{2}+2\sqrt{x^{2}+24}x^{3}-4x^{4}-96x^{2}+2\sqrt{x^{2}+24}x^{3}+48\sqrt{x^{2}+24}x+24\left(x^{2}-2x\sqrt{x^{2}+24}\right)+576$$

Combine -4\sqrt{x^{2}+24}x^{3} and 2\sqrt{x^{2}+24}x^{3} to get -2\sqrt{x^{2}+24}x^{3}.

$$4x^{4}-2\sqrt{x^{2}+24}x^{3}+72x^{2}-4x^{4}-96x^{2}+2\sqrt{x^{2}+24}x^{3}+48\sqrt{x^{2}+24}x+24\left(x^{2}-2x\sqrt{x^{2}+24}\right)+576$$

Combine 4x^{4} and -4x^{4} to get 0.

$$-2\sqrt{x^{2}+24}x^{3}+72x^{2}-96x^{2}+2\sqrt{x^{2}+24}x^{3}+48\sqrt{x^{2}+24}x+24\left(x^{2}-2x\sqrt{x^{2}+24}\right)+576$$

Combine 72x^{2} and -96x^{2} to get -24x^{2}.

$$-2\sqrt{x^{2}+24}x^{3}-24x^{2}+2\sqrt{x^{2}+24}x^{3}+48\sqrt{x^{2}+24}x+24\left(x^{2}-2x\sqrt{x^{2}+24}\right)+576$$

Combine -2\sqrt{x^{2}+24}x^{3} and 2\sqrt{x^{2}+24}x^{3} to get 0.

$$-24x^{2}+48\sqrt{x^{2}+24}x+24\left(x^{2}-2x\sqrt{x^{2}+24}\right)+576$$

Use the distributive property to multiply 24 by x^{2}-2x\sqrt{x^{2}+24}.

$$-24x^{2}+48\sqrt{x^{2}+24}x+24x^{2}-48\sqrt{x^{2}+24}x+576$$

Combine -24x^{2} and 24x^{2} to get 0.

$$48\sqrt{x^{2}+24}x-48\sqrt{x^{2}+24}x+576$$

Combine 48\sqrt{x^{2}+24}x and -48\sqrt{x^{2}+24}x to get 0.

$$576$$

Factor

$$2^{6}\times 3^{2}$$