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

To simplify the expression $(2x+3y)^2$, we use the algebraic identity for squaring a binomial.

Step 1: Identify the form of the expression. The expression $(2x+3y)^2$ is in the form of $(a+b)^2$. Here, $a = 2x$ and $b = 3y$.

Step 2: Recall the algebraic identity for $(a+b)^2$. The identity is: $(a+b)^2 = a^2 + 2ab + b^2$

Step 3: Substitute the values of $a$ and $b$ into the identity. Substitute $a=2x$ and $b=3y$ into the identity: $(2x+3y)^2 = (2x)^2 + 2(2x)(3y) + (3y)^2$

Step 4: Simplify each term. • Simplify the first term, $(2x)^2$: $(2x)^2 = 2^2 \cdot x^2 = 4x^2$ • Simplify the second term, $2(2x)(3y)$: Multiply the numerical coefficients and the variables: $2(2x)(3y) = (2 \cdot 2 \cdot 3)(x \cdot y) = 12xy$ 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, $(3y)^2$: $(3y)^2 = 3^2 \cdot y^2 = 9y^2$

Step 5: Combine the simplified terms. Substitute the simplified terms back into the expression: $(2x+3y)^2 = 4x^2 + 12xy + 9y^2$

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: $\boxed{4x^2 + 12xy + 9y^2}$