1. **State the problem:** Solve the equation $$\left(2+x\right)^2 + \left(\frac{20}{x} + 10\right)^2 = \left(40 \sqrt{x^2 + 10^2} + \sqrt{4 + \left(\frac{20}{x}\right)^2}\right)^2.$$\n\n2. **Rewrite and analyze the equation:** The equation involves squares and square roots. We want to simplify and isolate terms carefully.\n\n3. **Simplify inside the square roots:** Note that $$\sqrt{x^2 + 10^2} = \sqrt{x^2 + 100}$$ and $$\sqrt{4 + \left(\frac{20}{x}\right)^2} = \sqrt{4 + \frac{400}{x^2}}.$$\n\n4. **Expand the left side:** \n$$\left(2+x\right)^2 = x^2 + 4x + 4,$$\n$$\left(\frac{20}{x} + 10\right)^2 = \left(\frac{20}{x}\right)^2 + 2 \cdot \frac{20}{x} \cdot 10 + 10^2 = \frac{400}{x^2} + \frac{400}{x} + 100.$$\n\nSo the left side is $$x^2 + 4x + 4 + \frac{400}{x^2} + \frac{400}{x} + 100 = x^2 + 4x + \frac{400}{x^2} + \frac{400}{x} + 104.$$\n\n5. **Rewrite the right side:** \n$$\left(40 \sqrt{x^2 + 100} + \sqrt{4 + \frac{400}{x^2}}\right)^2 = \left(40 \sqrt{x^2 + 100}\right)^2 + 2 \cdot 40 \sqrt{x^2 + 100} \cdot \sqrt{4 + \frac{400}{x^2}} + \left(\sqrt{4 + \frac{400}{x^2}}\right)^2.$$\n\nCalculate each term:\n$$\left(40 \sqrt{x^2 + 100}\right)^2 = 1600 (x^2 + 100) = 1600 x^2 + 160000,$$\n$$\left(\sqrt{4 + \frac{400}{x^2}}\right)^2 = 4 + \frac{400}{x^2}.$$\n\nSo the right side is $$1600 x^2 + 160000 + 2 \cdot 40 \sqrt{x^2 + 100} \cdot \sqrt{4 + \frac{400}{x^2}} + 4 + \frac{400}{x^2} = 1600 x^2 + 160004 + \frac{400}{x^2} + 80 \sqrt{(x^2 + 100) \left(4 + \frac{400}{x^2}\right)}.$$\n\n6. **Set the equation:** \n$$x^2 + 4x + \frac{400}{x^2} + \frac{400}{x} + 104 = 1600 x^2 + 160004 + \frac{400}{x^2} + 80 \sqrt{(x^2 + 100) \left(4 + \frac{400}{x^2}\right)}.$$\n\n7. **Subtract common terms:** Subtract $$\frac{400}{x^2}$$ from both sides:\n$$x^2 + 4x + \frac{400}{x} + 104 = 1600 x^2 + 160004 + 80 \sqrt{(x^2 + 100) \left(4 + \frac{400}{x^2}\right)}.$$\n\n8. **Bring all terms except the root to one side:**\n$$x^2 + 4x + \frac{400}{x} + 104 - 1600 x^2 - 160004 = 80 \sqrt{(x^2 + 100) \left(4 + \frac{400}{x^2}\right)}.$$\n\nSimplify left side:\n$$-1599 x^2 + 4x + \frac{400}{x} - 159900 = 80 \sqrt{(x^2 + 100) \left(4 + \frac{400}{x^2}\right)}.$$\n\n9. **Square both sides to eliminate the square root:**\n$$\left(-1599 x^2 + 4x + \frac{400}{x} - 159900\right)^2 = 6400 (x^2 + 100) \left(4 + \frac{400}{x^2}\right).$$\n\n10. **Simplify the right side inside the product:**\n$$ (x^2 + 100) \left(4 + \frac{400}{x^2}\right) = 4 x^2 + 400 + \frac{40000}{x^2} + 40000 \cdot \frac{100}{x^2} \text{ (recheck carefully)}.$$\nActually, multiply carefully:\n$$ (x^2 + 100) \left(4 + \frac{400}{x^2}\right) = x^2 \cdot 4 + x^2 \cdot \frac{400}{x^2} + 100 \cdot 4 + 100 \cdot \frac{400}{x^2} = 4 x^2 + 400 + 400 + \frac{40000}{x^2} = 4 x^2 + 800 + \frac{40000}{x^2}.$$\n\n11. **So right side is:**\n$$6400 \left(4 x^2 + 800 + \frac{40000}{x^2}\right) = 25600 x^2 + 5120000 + \frac{256000000}{x^2}.$$\n\n12. **The equation is now:**\n$$\left(-1599 x^2 + 4x + \frac{400}{x} - 159900\right)^2 = 25600 x^2 + 5120000 + \frac{256000000}{x^2}.$$\n\n13. **This is a complicated algebraic equation in $x$.** To solve it exactly, one would expand the left side, multiply through by $x^2$ to clear denominators, and solve the resulting polynomial equation.\n\n14. **Summary:** The problem reduces to solving a high-degree polynomial equation after clearing denominators and squaring. Numerical methods or graphing may be used to find approximate solutions.\n\n**Final answer:** The equation simplifies to $$\left(-1599 x^2 + 4x + \frac{400}{x} - 159900\right)^2 = 25600 x^2 + 5120000 + \frac{256000000}{x^2}.$$\nSolving this for $x$ requires further algebraic manipulation or numerical approximation.