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The problem asks: **How many solution(s) does the equation \( x^2 + 9x + 20 = 2 \sqrt{x^2 + 9x + 23} \) have?**

Let's solve this step by step:

### 1. Start with the given equation:
\[
x^2 + 9x + 20 = 2 \sqrt{x^2 + 9x + 23}
\]

### 2. Isolate the square root term:
First, divide both sides by 2 to get rid of the coefficient on the right side:
\[
\frac{x^2 + 9x + 20}{2} = \sqrt{x^2 + 9x + 23}
\]

### 3. Square both sides to eliminate the square root:
Squaring both sides:
\[
\left( \frac{x^2 + 9x + 20}{2} \right)^2 = x^2 + 9x + 23
\]

This simplifies to:
\[
\frac{(x^2 + 9x + 20)^2}{4} = x^2 + 9x + 23
\]

### 4. Expand and simplify:
Expanding the left-hand side:
\[
(x^2 + 9x + 20)^2 = x^4 + 18x^3 + 109x^2 + 360x + 400
\]

So, the equation becomes:
\[
\frac{x^4 + 18x^3 + 109x^2 + 360x + 400}{4} = x^2 + 9x + 23
\]

### 5. Multiply both sides by 4 to eliminate the denominator:
\[
x^4 + 18x^3 + 109x^2 + 360x + 400 = 4(x^2 + 9x + 23)
\]

Expanding the right-hand side:
\[
x^4 + 18x^3 + 109x^2 + 360x + 400 = 4x^2 + 36x + 92
\]

### 6. Bring all terms to one side:
\[
x^4 + 18x^3 + 109x^2 + 360x + 400 - 4x^2 - 36x - 92 = 0
\]
Simplify:
\[
x^4 + 18x^3 + 105x^2 + 324x + 308 = 0
\]

### 7. Analyze the degree:
This is a quartic (degree 4) equation, so it can have up to 4 real solutions. However, the actual number of solutions can be fewer, depending on the discriminant and the behavior of the roots.

Thus, the equation could potentially have **4 solutions**.

The correct answer is:

**Option 4: 4 solutions**.

Would you like more details on solving quartic equations, or do you have any questions? 

Here are 5 related questions you can explore:
1. How do you solve a quartic equation step-by-step?
2. What are the conditions for a quartic equation to have real or complex solutions?
3. How does squaring both sides of an equation potentially introduce extraneous solutions?
4. What methods exist for solving higher-degree polynomial equations?
5. What is the significance of isolating a square root before squaring both sides?

**Tip:** Always check for extraneous solutions after squaring both sides, as they can arise from the squaring process.

How many solution(s) does the equation x^2 + 9x + 20 = 2√(x^2 + 9x + 23) have?

Learn how to solve the radical equation x^2 + 9x + 20 = 2√(x^2 + 9x + 23) with a detailed step-by-step solution. This problem covers quartic equations, radical manipulation, and solving higher-degree polynomials. Suitable for Grades 11-12.

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