Expanding the expression, we get:

(a+3c)^2 = a^2 + 2ac + 9c^2

(b+3c)(b-3c) = b^2 - 9c^2

Adding them together, we get:

(a+3c)^2 + (b+3c)(b-3c) = a^2 + 2ac + 9c^2 + b^2 - 9c^2

Simplifying further, we get the final expression:

a^2 + 2ac + 9c^2 + b^2 - 9c^2 = a^2 + b^2 + 2ac