To simplify the expression (m^2 + 6mn + 9n^2) ÷ (4m^2 + 12mn), we can first factor out a common factor from both the numerator and the denominator.

Factor out a common factor from the numerator:
m^2 + 6mn + 9n^2 = (m + 3n)(m + 3n) = (m + 3n)^2

Factor out a common factor from the denominator:
4m^2 + 12mn = 4(m^2 + 3mn) = 4m(m + 3n)

Now, the expression becomes:
(m + 3n)^2 / 4m(m + 3n)

Since (m + 3n) is common to both the numerator and the denominator, we can cancel them out:

= (m + 3n) / 4m

Therefore, the simplified form of the expression (m^2 + 6mn + 9n^2) ÷ (4m^2 + 12mn) is (m + 3n) / 4m.