Question

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

Answer 1

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