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 options $$\frac{b}{h}, \frac{b}{k}, \frac{45}{h}, \frac{45}{k}$$ must be an integer. 2. **Write the factorization and expand:** $$(hx + k)(x + j) = hx^2 + hjx + kx + kj = hx^2 + (hj + k)x + kj$$ 3. **Match coefficients with the original expression:** Since the original expression 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 $$j$$ are integer factors of $$-45$$. 5. **Analyze the coefficient of $$x^2$$:** We have $$h = 4$$, which is an integer. 6. **Check each option:** - (A) $$\frac{b}{h} = \frac{hj + k}{h} = j + \frac{k}{h}$$. Since $$k$$ and $$h=4$$ are integers, $$\frac{k}{4}$$ may not be an integer (e.g., if $$k=3$$, $$\frac{3}{4}$$ is not integer). So $$\frac{b}{h}$$ is not necessarily an integer. - (B) $$\frac{b}{k} = \frac{hj + k}{k} = \frac{hj}{k} + 1 = h \frac{j}{k} + 1$$. Since $$j$$ and $$k$$ are integers but $$\frac{j}{k}$$ may not be integer, $$\frac{b}{k}$$ is not necessarily integer. - (C) $$\frac{45}{h} = \frac{45}{4}$$. Since 4 does not divide 45 evenly, $$\frac{45}{h}$$ is not an integer. - (D) $$\frac{45}{k}$$. Since $$k$$ divides $$kj = -45$$, $$k$$ must be a divisor of 45. Therefore, $$\frac{45}{k}$$ is an integer. **Final answer:** Option (D) $$\frac{45}{k}$$ must be an integer.