jmc

If $4x^2-12x+a$ is the square of a binomial, then the binomial has the form $2x+b$ for some number $b$, since $(2x{)}^2=4x^2$. So, we compare $(2x+b{)}^2$ to $4x^2-12x+a$. Expanding $(2x+b{)}^2$ gives $$(2x+b{)}^2=4x^2+4bx+b^2.$$Equating the linear term of this to the linear term of $4x^2-12x+a$, we have $4bx=-12x$, so $b=-3$. Thus, $a=b^2=\boxed{9}$.