To solve this expression, we follow the order of operations (PEMDAS/BODMAS):

First, we simplify the expression inside the parentheses:
7/2 + 1/4 = (14/4) + (1/4) = 15/4

Next, we square the simplified expression:
(15/4)^2 = (15^2)/(4^2) = 225/16

Finally, we multiply the result by 4/5:
(4/5) * (225/16) = (4 * 225) / (5 * 16) = 900/80 = 45/4

Therefore, 4/5 * (7/2 + 1/4)^2 = 45/4.