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) \( \frac{45}{k} \)

To solve the expression \( 4 x^{2}+b x-45 \) and rewrite it in the form \( (h x+k)(x+j) \), we start by matching coefficients. If we assume \( h = 4 \), then we can express the terms inside the parentheses as follows: \[ (4x + k)(x + j) = 4x^2 + (4j + k)x + kj \] From this, we can identify that: - \( 4j + k = b \) - \( kj = -45 \) Now, \( k \) and \( j \) must be two integers whose product is \(-45\). The pairs of integers that multiply to give \(-45\) include \((1, -45), (-1, 45), (3, -15), (-3, 15), (5, -9), (-5, 9)\), and their reverses. To analyze which of the options must be an integer, we note that \( k \) varies depending on \( j \) and their multiplies. Since \( kj = -45 \), the value of \( \frac{45}{k} \) becomes critical. Using \( k \) from \( kj = -45 \), we can express this as \( \frac{45}{k} = -j \) which clearly must also be an integer since \( j \) is an integer. Thus, the correct answer is: (D) \( \frac{45}{k} \) must be an integer.