Replace ∗ with a monomial so that the trinomial may be represented by a square of a binomial: 25a^2 + ∗ + 1/4b^2

The square of a binomial is a pattern you would do well to remember.

... (x + y) ^2 = x^2 + 2xy + y^2

Your expression has x^2 = 25a^2, so x = 5a. It also has y^2 = (b^2) / 4, so y = b/2.

then the middle term is

... 2xy = 2 (5a) (b/2) = 5ab

The trinomial is

... 25a^2 + 5ab + (1/4) b^2 = (5a + b/2) ^2