Solve this and give me detailed analysis on how you came about the solutions

The expression $(2x+3y)^2$ is in the form of $(a+b)^2$.
Here, $a = 2x$ and $b = 3y$.

The identity is:

Substitute $a=2x$ and $b=3y$ into the identity:

• Simplify the first term, $(2x)^2$:

• Simplify the second term, $2(2x)(3y)$: Multiply the numerical coefficients and the variables:

This is where the $2ab$ in the previous step was an intermediate placeholder. The full simplification of $2(2x)(3y)$ results in $12xy$.

• Simplify the third term, $(3y)^2$:

Substitute the simplified terms back into the expression:

The detailed analysis of how the solution was derived is as follows:

1. We recognized the expression as a binomial squared, which immediately brings to mind the algebraic identity $(a+b)^2 = a^2 + 2ab + b^2$.
2. We identified the specific components of our expression that correspond to $a$ and $b$ in the identity. In this case, $a$ was $2x$ and $b$ was $3y$.
3. We systematically substituted these identified values of $a$ and $b$ into the identity. This resulted in $(2x)^2 + 2(2x)(3y) + (3y)^2$.
4. The next crucial step was to simplify each of these three terms individually.
   • For $(2x)^2$, we applied the power to both the coefficient and the variable, yielding $4x^2$.
   • For the middle term, $2(2x)(3y)$, we multiplied all the numerical coefficients together ($2 \times 2 \times 3 = 12$) and all the variables together ($x \times y = xy$). This resulted in $12xy$. This is the correct simplification of the $2ab$ part of the identity when $a=2x$ and $b=3y$.
   • For $(3y)^2$, similar to the first term, we applied the power to both the coefficient and the variable, yielding $9y^2$.
5. Finally, we combined these simplified terms to get the fully expanded and simplified form of the original expression.

The final simplified expression is: