Final answer:
The constant P must be 1 for the expression px² + 12x + 4p + px to be a perfect square. The expression as a perfect square would be (x + 2)². The conditions are matched and solved by setting the coefficients of the quadratic expression to form a perfect square trinomial.
Step-by-step explanation:
To find the possible value of the constant P for the expression to be a perfect square, we need to identify the conditions for the expression to be a perfect square, express it as such, and then calculate P. A perfect square trinomial is of the form (ax + b)2 = a2x2 + 2abx + b2. By comparison, the given expression px2 + (12 + p)x + 4p must match this form.
To express our given expression as a perfect square, we look for coefficients a and b such that a2 = p and b2 = 4p. The middle term suggests 2ab should equal 12 + p. Therefore, 2a2b should be equal to 2p3/2b since a2 = p and b = 2p1/2. So, 2p3/2b = 12+p. Solving for P from this equation, we find P must be equal to 1 for it to satisfy the perfect square condition as (sqrt(p)*2)^2 = 4p, which is our b2, and 2*sqrt(p)*2 = 4sqrt(p), which is indeed our middle term when P is 1.
Therefore, P is 1, and the expression is a perfect square when it is expressed as (x + 2)2 = x2 + 4x + 4. The middle term and constant term perfectly align for a perfect square trinomial fitting the correct formula.