15 The expression \( 4 x^{2}+b x-45 \), where \( b \) is a constant, can be rewritten as \( (h x+k)(x+j) \), where \( h, k \), and \( j \) are integer constants. Which of the following must be an integer? (A) \( \frac{b}{h} \) (B) \( \frac{b}{k} \) (C) \( \frac{45}{h} \) (D) \( \frac{45}{k} \)

We start with the expression   4x² + bx – 45 which is given to factor as   (hx + k)(x + j), with h, k, j being integers. Step 1. Expand the factored form:   (hx + k)(x + j) = hx² + (hj + k)x + kj Step 2. Match coefficients with 4x² + bx – 45:   Coefficient of x²: h = 4       (since 4x² = hx²)   Coefficient of x: hj + k = b   Constant term: kj = –45 Step 3. Observe that k and j are factors (divisors) of –45. This means k must be an integer divisor of –45. Consequently, k divides 45. Therefore, the expression   45/k is an integer since if k is a divisor of –45, then |k| is a divisor of 45. Now, check the other options briefly: (A) b/h = (4j + k)/4 = j + (k/4) need not be an integer unless k is a multiple of 4 (which is not guaranteed). (B) b/k = (4j + k)/k = 4j/k + 1 might not be an integer unless k divides 4j. (C) 45/h = 45/4 is not an integer because 4 does not evenly divide 45. Only (D) ensures an integer value. Thus, the answer is (D) 45/k.