Short Answer 
The process for factoring the quadratic expression 4x¬≤ + bx Р45 involves recognizing it as the product of two binomials, identifying integer pairs (k, j) whose product equals -45, and ensuring that 45/k yields an integer. By checking the factor pairs of -45, any suitable k will allow for this condition to be met, leading to the conclusion that Option D is the correct answer.
Step 1: Understand the Quadratic Factorization
To solve the given quadratic expression, recognize that it can be rewritten in the form of two binomials, which generally looks like (hx + k)(x + j). In this specific case, we have to work with the expression 4x¬≤ + bx Р45. Our goal is to factor it properly, which will help us find the values of k and j.
Step 2: Identify the Conditions for Integer Values
Since the constant term is -45, it is crucial to remember that k and j must both be integers. The product of these constants (k ‚àöo j) should equal -45. Hence, we need to check the factors of -45 to see which combinations of k and j yield integer values. List the possible pairs of factors:
- (1, -45)
- (-1, 45)
- (3, -15)
- (-3, 15)
- (5, -9)
- (-5, 9)
Step 3: Find the Required Integer Value
For our expression to hold true, we must ensure that the fraction 45/k also results in an integer. Thus, we must determine which k from our pairs can divide 45 without leaving a remainder. As k takes integer values from our factor pairs, any valid selection will lead to 45/k being an integer. Finally, based on this understanding, the correct answer to the problem is Option D.