1. **Problem Statement:** We have the quadratic expression $4x^2 + bx - 45$, where $b$ is a constant. It can be factored as $(hx + k)(x + j)$, where $h$, $k$, and $j$ are integers. We need to determine which of the options $\frac{b}{h}$, $\frac{b}{k}$, $\frac{45}{h}$, or $\frac{45}{k}$ must be an integer. 2. **Factoring Setup:** The expression is factored as $(hx + k)(x + j)$. Expanding this gives: $$ (hx + k)(x + j) = hx^2 + hjx + kx + kj = hx^2 + (hj + k)x + kj $$ 3. **Matching Coefficients:** Comparing with $4x^2 + bx - 45$, we get: - Coefficient of $x^2$: $h = 4$ - Coefficient of $x$: $b = hj + k$ - Constant term: $kj = -45$ 4. **Analyze the constant term:** Since $k$ and $j$ are integers and $kj = -45$, $k$ must be a divisor of $-45$. Therefore, $k$ divides $-45$ and so $\frac{45}{k}$ is an integer. 5. **Analyze the other options:** - $b/h = \frac{hj + k}{h} = j + \frac{k}{h}$. Since $h=4$ and $k$ is an integer divisor of $-45$, $\frac{k}{4}$ may not be an integer. - $b/k = \frac{hj + k}{k} = h \frac{j}{k} + 1$. Since $j$ and $k$ are integers, $\frac{j}{k}$ may not be an integer. - $\frac{45}{h} = \frac{45}{4}$ is not necessarily an integer. 6. **Conclusion:** The only quantity that must be an integer is $\frac{45}{k}$. **Final answer:** D) $\frac{45}{k}$ must be an integer.