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The question in the image is asking about the function \( f(x) = \frac{ax + 5}{x^2 + bx + 1} \), where \( a > 0 \) and \( b < 0 \), and the graph of \( f \) has one vertical asymptote and one horizontal asymptote given by \( y = -3 \). The goal is to find the value of \( a + 2b \).

### Solution outline:

1. **Vertical Asymptote**: A vertical asymptote occurs when the denominator of the function equals zero. So, we need to solve \( x^2 + bx + 1 = 0 \). Since the problem states there is only **one** vertical asymptote, the quadratic must have exactly **one root**, which implies a **double root**. This happens when the discriminant is zero.

   The discriminant \( \Delta \) for the quadratic \( x^2 + bx + 1 = 0 \) is:
   \[
   \Delta = b^2 - 4(1)(1) = b^2 - 4
   \]
   For there to be one vertical asymptote (double root), we set \( \Delta = 0 \):
   \[
   b^2 - 4 = 0 \quad \Rightarrow \quad b^2 = 4 \quad \Rightarrow \quad b = -2 \, \text{(since \( b < 0 \))}.
   \]

2. **Horizontal Asymptote**: The horizontal asymptote is determined by the degrees of the numerator and the denominator. Since both are of degree 1, the horizontal asymptote is given by the ratio of the leading coefficients:
   \[
   \text{Horizontal asymptote}: y = \frac{a}{1}
   \]
   According to the problem, this asymptote is \( y = -3 \), so we have:
   \[
   \frac{a}{1} = -3 \quad \Rightarrow \quad a = -3
   \]

3. **Calculate \( a + 2b \)**: 
   Now that we know \( a = -3 \) and \( b = -2 \), we can find:
   \[
   a + 2b = -3 + 2(-2) = -3 - 4 = -7
   \]

However, since the possible answers are -2, -1, 1, and 2, it seems the options may not align with this exact result, which suggests there could be a different interpretation or possible typo in the problem setup. 

If you have more details or corrections, feel free to share!

Would you like further details or clarification on any step? Here are some related questions to explore:

1. How are horizontal and vertical asymptotes determined in rational functions?
2. What is the significance of the discriminant in quadratic equations?
3. How do vertical and horizontal asymptotes affect the shape of a graph?
4. What happens when a function has a double root at the vertical asymptote?
5. Why does \( a > 0 \) and \( b < 0 \) matter in the given function?

**Tip:** Always check for the degree of the numerator and denominator when determining horizontal asymptotes in rational functions!

Diketahui fungsi f(x) = (ax + 5)/(x^2 + bx + 1) dengan a > 0 dan b < 0. Jika grafik f mempunyai satu asimtot tegak dan salah satu asimtot datarnya adalah y = -3, maka a + 2b adalah?

Learn how to find vertical and horizontal asymptotes for a rational function f(x) = (ax + 5)/(x^2 + bx + 1) where a > 0 and b < 0, and solve for a + 2b given that one asymptote is y = -3.

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