Expand and Simplify: (3y+4a)² - 9(y-2a)(y+2a)

To solve the problem, we will follow these steps:

• Step 1: Expand the square of the binomial $(3y + 4a)$.
• Step 2: Simplify the product of the binomials $9(y-2a)(y+2a)$.
• Step 3: Subtract the second result from the first and simplify.

Let's work through each step:
Step 1: The expression $(3y + 4a)^2$ can be expanded as follows:

$(3y + 4a)^2 = (3y)^2 + 2(3y)(4a) + (4a)^2 = 9y^2 + 24ay + 16a^2$.

Step 2: Simplify the expression $9(y-2a)(y+2a)$. This uses the difference of squares formula where $(y-2a)(y+2a) = y^2 - (2a)^2$:

$(y-2a)(y+2a) = y^2 - 4a^2$.

Then multiply by 9:

$9(y^2 - 4a^2) = 9y^2 - 36a^2$.

Step 3: Now subtract the second result from the first:

$(9y^2 + 24ay + 16a^2) - (9y^2 - 36a^2) = 9y^2 + 24ay + 16a^2 - 9y^2 + 36a^2$.

Combine and simplify:

$24ay + 52a^2$.

Factoring out $4a$ from the expression:

$4a(6y + 13a)$.

Therefore, the solution to the problem is: $4a(6y + 13a)$.