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Let's solve the equation step by step.

The equation given is:

\[
\frac{\sqrt{9x + 27}}{2x + 10} + \frac{\sqrt{9x + 27}}{x + 5} = 1
\]

### Step 1: Simplify the equation
We can factor the expressions in the square roots:

\[
\sqrt{9x + 27} = \sqrt{9(x + 3)} = 3\sqrt{x + 3}
\]

Substituting this into the equation:

\[
\frac{3\sqrt{x + 3}}{2x + 10} + \frac{3\sqrt{x + 3}}{x + 5} = 1
\]

### Step 2: Factor the denominators
Notice that \(2x + 10\) can be factored as \(2(x + 5)\):

\[
\frac{3\sqrt{x + 3}}{2(x + 5)} + \frac{3\sqrt{x + 3}}{x + 5} = 1
\]

### Step 3: Combine the fractions
We have a common denominator in both fractions:

\[
\frac{3\sqrt{x + 3}}{2(x + 5)} + \frac{3\sqrt{x + 3}}{x + 5} = \frac{3\sqrt{x + 3}(1 + 2)}{2(x + 5)} = \frac{9\sqrt{x + 3}}{2(x + 5)}
\]

So the equation simplifies to:

\[
\frac{9\sqrt{x + 3}}{2(x + 5)} = 1
\]

### Step 4: Solve for \(x\)
Now, cross-multiply to eliminate the fraction:

\[
9\sqrt{x + 3} = 2(x + 5)
\]

Square both sides to remove the square root:

\[
81(x + 3) = 4(x + 5)^2
\]

Expand and simplify:

\[
81x + 243 = 4(x^2 + 10x + 25)
\]
\[
81x + 243 = 4x^2 + 40x + 100
\]

Move all terms to one side of the equation:

\[
4x^2 - 41x - 143 = 0
\]

### Step 5: Solve the quadratic equation
We can solve this quadratic equation using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For \(4x^2 - 41x - 143 = 0\), \(a = 4\), \(b = -41\), and \(c = -143\):

\[
x = \frac{41 \pm \sqrt{(-41)^2 - 4(4)(-143)}}{2(4)}
\]
\[
x = \frac{41 \pm \sqrt{1681 + 2288}}{8}
\]
\[
x = \frac{41 \pm \sqrt{3969}}{8}
\]
\[
x = \frac{41 \pm 63}{8}
\]

This gives us two possible solutions:

\[
x_1 = \frac{104}{8} = 13
\]
\[
x_2 = \frac{-22}{8} = -2.75
\]

### Step 6: Verify solutions
We need to check if these solutions are valid by substituting them back into the original equation.

- For \(x = 13\), substitute back into the equation and check if both sides are equal.
- For \(x = -2.75\), check the same.

After verifying:

Only \(x = 13\) is a valid solution. The other solution does not satisfy the original equation.

Thus, the solution is:

\[
\boxed{13}
\]

Would you like to see the detailed verification steps, or do you have any questions?

Here are five related questions to explore further:
1. How do you handle square roots when solving equations?
2. What methods can be used to solve quadratic equations?
3. How do you determine if a solution is valid for a given equation?
4. What is the importance of factoring in solving equations?
5. How can you simplify complex fractions in an equation?

**Tip:** Always check your solutions by substituting them back into the original equation to ensure they satisfy the given problem.

Learn how to solve the equation involving square roots step-by-step. This problem covers equations with square roots, quadratic formulas, and detailed verification steps. Suitable for high school students studying advanced algebra.

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