To solve this problem, let's break it down:

1. Given Information:

   • The ratio of the length to the width of the rectangle is $3:2$.
   • The area of the rectangle is 12.

2. Define the Variables:

   • Let the width of the rectangle be $2x$ and the length be $3x$, where $x$ is a scaling factor.

3. Area of a Rectangle:

   • The area of a rectangle is given by $\text{Area} = \text{Length} \times \text{Width}$.

   Using the variables we defined, the area becomes: $\text{Area} = (3x)(2x) = 6x^2$

4. Set the Equation for Area: Since the area is given as 12, the equation becomes: $6x^2 = 12$

This simplifies to $x^2 = 2$, which allows us to find the dimensions of the rectangle.

Now, analyzing the given options:

• Option 1: $(3x) + (2x) = 12$

  • This adds the length and width, but that is not how area is calculated. This equation is incorrect.

• Option 2: $(3x)(2x) = 12$

  • This represents the correct way to calculate the area. This is the correct equation.

• Option 3: $(3x) + (3x) + (2x) + (2x) = 12$

  • This adds all the sides, which is the perimeter, not the area. This equation is incorrect.

• Option 4: $(3x)^2 + (2x)^2 = 12$

  • This sums the squares of the length and width, which does not represent area. This equation is incorrect.

Conclusion:

The correct equation is Option 2: $(3x)(2x) = 12$.

Would you like more details or have any questions? Here are some follow-up questions you might find helpful:

1. How do you solve for $x$ in the equation $6x^2 = 12$?
2. Once $x$ is found, how do you calculate the dimensions of the rectangle?
3. Can you generalize how to approach problems involving ratios of length and width?
4. How would you find the perimeter of the rectangle?
5. What are other ways to represent ratios in geometry problems?

Tip: Always remember that the area of a rectangle is the product of its length and width, not their sum.