Final answer:
To determine the values of a and p, we can expand and simplify the given expression (a + 3i)(5 - i). By combining like terms and setting the imaginary part equal to zero, we can solve for a and find that a = 15i. Substituting this value back into the expression, we find that p = 78i.
Step-by-step explanation:
The product of two complex numbers can be found by using the distributive property. In this case, we have (a + 3i)(5 - i). Multiplying these two complex numbers together gives us:
(a + 3i)(5 - i) = 5a + 15i - ai - 3i²
Since i² = -1, we can simplify further:
(a + 3i)(5 - i) = 5a + 15i - ai - 3(-1) = 5a + 15i - ai + 3 = 5a + 15i + 3 - ai = (5a + 3) + (15i - a)
Since p is a real number, the imaginary part must be zero. Therefore, we have:
15i - a = 0
a = 15i
Substituting this value of a back into the expression for p:
p = (5a + 3) + (15i - a) = (5(15i) + 3) + (15i - 15i) = 75i + 3i = 78i