1. **State the problem:** We have the quadratic expression $$4x^2 + bx - 45$$ where $$b$$ is a constant. It can be factored as $$(hx + k)(x + j)$$ where $$h, k, j$$ are integers. We need to determine which of the following must be an integer: $$\frac{b}{h}$$, $$\frac{b}{k}$$, $$\frac{45}{h}$$, or $$\frac{45}{k}$$. 2. **Write the factored form and expand:** $$ (hx + k)(x + j) = hx^2 + hjx + kx + kj = hx^2 + (hj + k)x + kj $$ 3. **Match coefficients with the original quadratic:** Since the original quadratic is $$4x^2 + bx - 45$$, we have: - 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 $$kj = -45$$, both $$k$$ and $$j$$ must be integer factors of $$-45$$. 5. **Analyze the coefficient of $$x^2$$:** We have $$h = 4$$, which is an integer. 6. **Analyze the coefficient of $$x$$:** $$b = hj + k = 4j + k$$, where $$j$$ and $$k$$ are integers. 7. **Check each option:** - (A) $$\frac{b}{h} = \frac{4j + k}{4}$$. Since $$k$$ may not be divisible by 4, this is not necessarily an integer. - (B) $$\frac{b}{k} = \frac{4j + k}{k} = 4\frac{j}{k} + 1$$. Since $$j/k$$ is not necessarily an integer, this is not necessarily an integer. - (C) $$\frac{45}{h} = \frac{45}{4}$$. Since 4 does not divide 45 evenly, this is not an integer. - (D) $$\frac{45}{k}$$. Since $$k$$ divides $$-45$$ (because $$kj = -45$$), $$k$$ is a divisor of 45, so $$\frac{45}{k}$$ is an integer. **Final answer:** Option D, $$\frac{45}{k}$$ must be an integer.