Evaluate the Expression: What is (a+3b)²-(3b-a)²?

$(a+3b)^2-(3b-a)^2=\text{?}$

To solve this problem, we will follow these steps:

• Step 1: Expand $(a+3b)^2$.
• Step 2: Expand $(3b-a)^2$.
• Step 3: Subtract the result of step 2 from step 1.

Now, let's work through the calculations:

Step 1: Expand $(a+3b)^2$
Using the formula $(x + y)^2 = x^2 + 2xy + y^2$, we let $x = a$ and $y = 3b$ to get:
$(a+3b)^2 = a^2 + 2 \cdot a \cdot 3b + (3b)^2$
$= a^2 + 6ab + 9b^2$.

Step 2: Expand $(3b-a)^2$
Again using the squaring formula, letting $x = 3b$ and $y = -a$, we have:
$(3b-a)^2 = (3b)^2 - 2 \cdot 3b \cdot a + a^2$
$= 9b^2 - 6ab + a^2$.

Step 3: Perform the subtraction
We subtract the expansion of $(3b-a)^2$ from $(a+3b)^2$:
$(a^2 + 6ab + 9b^2) - (9b^2 - 6ab + a^2)$
$= a^2 + 6ab + 9b^2 - 9b^2 + 6ab - a^2$
= $12ab$.

The solution to the problem is $12ab$, which corresponds to choice 2.