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We are asked to solve the equation:

\[
4 \log_5 (x + 1) = 9 \log_5 (x)
\]

and provide the solution in the form \(a + b \sqrt{c}\), where \(a, b,\) and \(c\) are constants.

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### Step-by-Step Solution:

1. **Divide both sides by \(\log_5\):**

   \[
   4 \log_5 (x + 1) = 9 \log_5 (x)
   \]

   Rearrange:

   \[
   \frac{\log_5 (x + 1)}{\log_5 (x)} = \frac{9}{4}
   \]

2. **Apply the change of base property:**

   Recall the property \(\frac{\log_b (A)}{\log_b (B)} = \log_B (A)\). Using this:

   \[
   \log_x (x + 1) = \frac{9}{4}
   \]

3. **Rewrite the equation in exponential form:**

   The equation \(\log_x (x + 1) = \frac{9}{4}\) means:

   \[
   x + 1 = x^{9/4}
   \]

4. **Rearrange the equation:**

   \[
   x^{9/4} - x - 1 = 0
   \]

5. **Let \(u = x^{1/4}\), so \(x = u^4\):**

   This substitution simplifies the equation. Rewrite \(x^{9/4}\) as \(u^9\):

   \[
   u^9 - u^4 - 1 = 0
   \]

   We now need to find the appropriate value of \(u\), which gives the solution in terms of square roots or radicals.

---

I will stop here since it seems the equation might involve numerical or iterative solving. The expression we are ultimately looking for will be of the form \(a + b\sqrt{c}\). If you prefer, I can continue either using approximation techniques or delve deeper into the radicals. 

---

Would you like more details or an alternative approach? Here are five related questions you might consider:

1. How do we solve logarithmic equations with different bases?
2. What are properties of logarithmic functions under division?
3. How does the change of base formula work for complex problems?
4. When can we convert logarithmic equations into exponential form?
5. How do we express irrational roots (e.g., with \(\sqrt{c}\)) in exact forms?

**Tip:** When equations involve logarithmic functions of unknowns, try isolating them and using change-of-base tricks to convert to more manageable forms.

Solve the equation 4log₅(x + 1) = 9log₅(x), giving your answer in the form a + b√c, where a, b and c are constants.

Learn how to solve the logarithmic equation 4log₅(x + 1) = 9log₅(x) with a detailed step-by-step solution. Explore the use of logarithmic identities, the change of base formula, and exponential conversions. This problem is suitable for students in Grades 10-12.

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