https://file.aiquickdraw.com/mathbot/file/1729260416118z2ufd17n.jpg

Let's analyze and solve the equation given in the image. The equation appears to be:

\[
\frac{x}{x+8} - \frac{5}{x+9} = \frac{-8x - 72}{x^2 + 17x + 72}
\]

### Step-by-Step Solution:

1. **Factor the denominator on the right-hand side**:  
   The quadratic denominator can be factored as:
   \[
   x^2 + 17x + 72 = (x+8)(x+9)
   \]
   So, the equation becomes:
   \[
   \frac{x}{x+8} - \frac{5}{x+9} = \frac{-8x - 72}{(x+8)(x+9)}
   \]

2. **Find a common denominator on the left-hand side**:  
   To combine the left-hand side fractions, we need to have a common denominator, which is \((x+8)(x+9)\):
   \[
   \frac{x}{x+8} = \frac{x(x+9)}{(x+8)(x+9)} = \frac{x^2 + 9x}{(x+8)(x+9)}
   \]
   \[
   \frac{5}{x+9} = \frac{5(x+8)}{(x+8)(x+9)} = \frac{5x + 40}{(x+8)(x+9)}
   \]
   Now, we can rewrite the left-hand side as:
   \[
   \frac{x^2 + 9x}{(x+8)(x+9)} - \frac{5x + 40}{(x+8)(x+9)} = \frac{(x^2 + 9x) - (5x + 40)}{(x+8)(x+9)}
   \]
   Simplifying the numerator:
   \[
   (x^2 + 9x) - (5x + 40) = x^2 + 9x - 5x - 40 = x^2 + 4x - 40
   \]
   So the equation becomes:
   \[
   \frac{x^2 + 4x - 40}{(x+8)(x+9)} = \frac{-8x - 72}{(x+8)(x+9)}
   \]

3. **Eliminate the denominators**:  
   Since both sides of the equation now have the same denominator, we can cancel out \((x+8)(x+9)\) from both sides:
   \[
   x^2 + 4x - 40 = -8x - 72
   \]

4. **Move all terms to one side**:
   \[
   x^2 + 4x - 40 + 8x + 72 = 0
   \]
   Simplifying:
   \[
   x^2 + 12x + 32 = 0
   \]

5. **Solve the quadratic equation**:  
   We now solve the quadratic equation \(x^2 + 12x + 32 = 0\) using the quadratic formula:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   where \(a = 1\), \(b = 12\), and \(c = 32\). Substituting into the formula:
   \[
   x = \frac{-12 \pm \sqrt{12^2 - 4(1)(32)}}{2(1)} = \frac{-12 \pm \sqrt{144 - 128}}{2} = \frac{-12 \pm \sqrt{16}}{2}
   \]
   \[
   x = \frac{-12 \pm 4}{2}
   \]
   So, the two possible solutions are:
   \[
   x = \frac{-12 + 4}{2} = \frac{-8}{2} = -4
   \]
   \[
   x = \frac{-12 - 4}{2} = \frac{-16}{2} = -8
   \]

6. **Check for extraneous solutions**:  
   Since \(x = -8\) would make the denominators zero, this is an extraneous solution. Therefore, the only valid solution is:
   \[
   x = -4
   \]

### Final Answer:
\[
x = -4
\]

Would you like further details or clarification on any part of this solution?

---

**Related Questions:**
1. How do we factor quadratic equations like \(x^2 + 17x + 72\)?
2. Why do we need to check for extraneous solutions in rational equations?
3. How does the quadratic formula work and when should it be used?
4. What are some common mistakes to avoid when solving rational equations?
5. Can you explain other methods for solving quadratics apart from the quadratic formula?

**Tip:** Always check for restrictions in rational equations to avoid invalid solutions that make the denominator zero.

Solve for x: \(\frac{x}{x+8} - \frac{5}{x+9} = \frac{-8x - 72}{x^2 + 17x + 72}\)

Learn how to solve the rational equation \(\frac{x}{x+8} - \frac{5}{x+9} = \frac{-8x - 72}{x^2 + 17x + 72}\). This step-by-step guide walks through factoring the quadratic, finding common denominators, and using the quadratic formula to solve for x. Suitable for students in Grades 9-12.

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