1. **State the problem:** Find the exact values of $x$ such that $$21\log_5(x + 5) - \log_5(2x + 2) = 2.$$\n\n2. **Recall the logarithm properties:**\n- $a\log_b(c) = \log_b(c^a)$\n- $\log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right)$\n\n3. **Apply the properties:**\nRewrite the left side using the power rule:\n$$21\log_5(x + 5) = \log_5\left((x + 5)^{21}\right).$$\nSo the equation becomes:\n$$\log_5\left((x + 5)^{21}\right) - \log_5(2x + 2) = 2.$$\n\n4. Use the subtraction property of logs:\n$$\log_5\left(\frac{(x + 5)^{21}}{2x + 2}\right) = 2.$$\n\n5. Convert the logarithmic equation to exponential form:\n$$\frac{(x + 5)^{21}}{2x + 2} = 5^2 = 25.$$\n\n6. Multiply both sides by $2x + 2$:\n$$(x + 5)^{21} = 25(2x + 2).$$\n\n7. Factor $2x + 2$:\n$$2x + 2 = 2(x + 1).$$\nSo:\n$$(x + 5)^{21} = 50(x + 1).$$\n\n8. We look for solutions where both sides are equal. Since the left side is a very large power, the simplest way is to check if $x + 5$ and $x + 1$ relate nicely.\n\n9. Try to find $x$ such that $x + 5 = a$ and $x + 1 = b$ satisfy $a^{21} = 50b$.\n\n10. Note that $b = x + 1 = a - 4$. Substitute:\n$$a^{21} = 50(a - 4).$$\n\n11. This is a complicated equation to solve exactly, but since the problem asks for exact values as simplified surds, check if $a$ is a root of $a^{21} - 50a + 200 = 0$.\n\n12. Try $a = 5$:\n$$5^{21} - 50 \times 5 + 200 = 5^{21} - 250 + 200 = 5^{21} - 50,$$ which is huge and not zero.\n\n13. Try $a = 2$:\n$$2^{21} - 50 \times 2 + 200 = 2097152 - 100 + 200 = 2097252,$$ not zero.\n\n14. Since direct solving is impractical, consider the original equation again.\n\n15. Alternatively, rewrite the original equation as:\n$$21\log_5(x + 5) - \log_5(2x + 2) = 2,$$\nwhich can be rearranged to:\n$$\log_5\left(\frac{(x + 5)^{21}}{2x + 2}\right) = 2.$$\n\n16. Exponentiate both sides:\n$$\frac{(x + 5)^{21}}{2x + 2} = 25.$$\n\n17. Multiply both sides by $2x + 2$:\n$$(x + 5)^{21} = 25(2x + 2) = 50(x + 1).$$\n\n18. Since $x + 5 > 0$ and $2x + 2 > 0$ for the logs to be defined, $x > -5$ and $x > -1$. So domain is $x > -1$.\n\n19. Try to find $x$ such that $x + 5 = \sqrt[21]{50(x + 1)}$.\n\n20. Let $y = x + 5$, then $x + 1 = y - 4$, so:\n$$y = \sqrt[21]{50(y - 4)}.$$\n\n21. Raise both sides to the 21st power:\n$$y^{21} = 50(y - 4).$$\n\n22. Rearranged:\n$$y^{21} - 50y + 200 = 0.$$\n\n23. This is a polynomial equation of degree 21, which is difficult to solve exactly.\n\n24. Since the problem asks for exact values as simplified surds, check if $y = 5$ is a root:\n$$5^{21} - 50 \times 5 + 200 = 5^{21} - 250 + 200 = 5^{21} - 50,$$ not zero.\n\n25. Check $y = 2$:\n$$2^{21} - 100 + 200 = 2097152 + 100,$$ not zero.\n\n26. Since no simple integer root, the only exact solution is to express $x$ as:\n$$x = y - 5,$$ where $y$ satisfies $$y^{21} = 50(y - 4).$$\n\n27. **Final answer:** The exact values of $x$ satisfy $$ (x + 5)^{21} = 50(x + 1),$$ or equivalently $$x = y - 5$$ where $$y^{21} = 50(y - 4).$$\n\nThis implicit form is the exact solution.\n\n**Domain restriction:** $x > -1$ to keep the logarithms defined.