Simplify:
`(2x+3y)^(2)+(2x-3y)^(2)`

Answer

To simplify the expression \((2x + 3y)^2 + (2x - 3y)^2\), we can use the identities for the square of a binomial. Let's break it down step by step.

### Step 1: Apply the Binomial Square Formula
We know the formulas:
- \((A + B)^2 = A^2 + B^2 + 2AB\)
- \((A - B)^2 = A^2 + B^2 - 2AB\)

Here, let \(A = 2x\) and \(B = 3y\).

### Step 2: Calculate \((2x + 3y)^2\)
Using the first formula:
\[ (2x + 3y)^2 = (2x)^2 + (3y)^2 + 2(2x)(3y) \]
Calculating each term:
- \((2x)^2 = 4x^2\)
- \((3y)^2 = 9y^2\)
- \(2(2x)(3y) = 12xy\)

So,
\[ (2x + 3y)^2 = 4x^2 + 9y^2 + 12xy \]

### Step 3: Calculate \((2x - 3y)^2\)
Using the second formula:
\[ (2x - 3y)^2 = (2x)^2 + (3y)^2 - 2(2x)(3y) \]
Calculating each term:
- \((2x)^2 = 4x^2\)
- \((3y)^2 = 9y^2\)
- \(-2(2x)(3y) = -12xy\)

So,
\[ (2x - 3y)^2 = 4x^2 + 9y^2 - 12xy \]

### Step 4: Combine the Two Results
Now we add the two results together:
\[ (2x + 3y)^2 + (2x - 3y)^2 = (4x^2 + 9y^2 + 12xy) + (4x^2 + 9y^2 - 12xy) \]

### Step 5: Simplify the Expression
Combining like terms:
- The \(12xy\) and \(-12xy\) cancel each other out.
- The \(4x^2\) terms add up: \(4x^2 + 4x^2 = 8x^2\)
- The \(9y^2\) terms add up: \(9y^2 + 9y^2 = 18y^2\)

Thus, we have:
\[ (2x + 3y)^2 + (2x - 3y)^2 = 8x^2 + 18y^2 \]

### Final Answer
The simplified expression is:
\[ \boxed{8x^2 + 18y^2} \]