Answer:
p = 2 or p = -2 4/9
Explanation:
We can substitute x=3/2 into the equation and solve for the values of p that make the result be zero.
p^2(3/2)^2 -12(3/2) +p +7 = 0
9/4p^2 -18 + p + 7 = 0 . . . . eliminate parentheses
9p^2 +4p -44 = 0 . . . . . . simplify and multiply by 4
(9p +22)(p -2) = 0 . . . . . factor
Values of p that make 3/2 be one of the zeros are ...
p = -22/9, p = 2
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Additional comment
The zero product rule tells you a product will be zero if and only if a factor is zero. Hence the solutions to the quadratic are values of p that make the factors zero.
9p +22 = 0 ⇒ 9p = -22 ⇒ p = -22/9
p -2 = 0 ⇒ p = 2
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For p=2, the solution 3/2 has multiplicity 2. For p=-22/9, the other zero is x=123/242.