To expand $(3x + 4y)^2$, we apply the binomial expansion formula, which is:

$(a + b)^2 = a^2 + 2ab + b^2$

Here, $a = 3x$ and $b = 4y$. Let's expand:

$(3x + 4y)^2 = (3x)^2 + 2(3x)(4y) + (4y)^2$

Now, calculate each term:

1. $(3x)^2 = 9x^2$
2. $2(3x)(4y) = 24xy$
3. $(4y)^2 = 16y^2$

So the expansion is:

$9x^2 + 24xy + 16y^2$

Let me know if you want further details or have any questions!

Here are 5 related questions you might find interesting:

1. How do you expand $(2a + 5b)^2$?
2. What is the general form for $(ax + by)^2$?
3. How do you expand a binomial raised to the power of 3, such as $(x + y)^3$?
4. What happens if you expand $(3x - 4y)^2$?
5. How does the expansion change if there are coefficients in front of the binomial?

Tip: Binomial expansion is a powerful tool, especially when dealing with polynomials in algebraic manipulations!