Simpilfy `(2sqrt(5) + 3sqrt(2))^(2)`

Answer

To simplify \((2\sqrt{5} + 3\sqrt{2})^{2}\), we can follow these steps:

### Step 1: Identify the expression
We have the expression \((A + B)^{2}\) where:
- \(A = 2\sqrt{5}\)
- \(B = 3\sqrt{2}\)

### Step 2: Use the formula for squaring a binomial
The formula for squaring a binomial is:
\[ (A + B)^{2} = A^{2} + 2AB + B^{2} \]

### Step 3: Calculate \(A^{2}\)
Now, we calculate \(A^{2}\):
\[ A^{2} = (2\sqrt{5})^{2} = 2^{2} \cdot (\sqrt{5})^{2} = 4 \cdot 5 = 20 \]

### Step 4: Calculate \(B^{2}\)
Next, we calculate \(B^{2}\):
\[ B^{2} = (3\sqrt{2})^{2} = 3^{2} \cdot (\sqrt{2})^{2} = 9 \cdot 2 = 18 \]

### Step 5: Calculate \(2AB\)
Now, we calculate \(2AB\):
\[ 2AB = 2 \cdot (2\sqrt{5}) \cdot (3\sqrt{2}) = 2 \cdot 2 \cdot 3 \cdot \sqrt{5} \cdot \sqrt{2} = 12\sqrt{10} \]

### Step 6: Combine all parts
Now we combine all the parts together:
\[ (2\sqrt{5} + 3\sqrt{2})^{2} = A^{2} + 2AB + B^{2} = 20 + 12\sqrt{10} + 18 \]

### Step 7: Simplify the expression
Finally, we simplify the expression:
\[ 20 + 18 + 12\sqrt{10} = 38 + 12\sqrt{10} \]

### Final Answer
Thus, the simplified form of \((2\sqrt{5} + 3\sqrt{2})^{2}\) is:
\[ \boxed{38 + 12\sqrt{10}} \]