Simplify:
`(5x+3y+2)^(2)+(5x+3y-2)^(2)`

Answer

To simplify the expression \((5x + 3y + 2)^{2} + (5x + 3y - 2)^{2}\), we can follow these steps:

### Step 1: Expand both squares using the identity
We will use the identity \((a + b)^{2} = a^{2} + b^{2} + 2ab\).

Let \(a = 5x + 3y\) and \(b = 2\). Therefore, we can rewrite the expression as:
\[ (5x + 3y + 2)^{2} + (5x + 3y - 2)^{2} \]

### Step 2: Expand \((5x + 3y + 2)^{2}\)
Using the identity:
\[ (5x + 3y + 2)^{2} = (5x + 3y)^{2} + 2^{2} + 2(5x + 3y)(2) \]
Calculating each part:
- \((5x + 3y)^{2} = (5x)^{2} + (3y)^{2} + 2(5x)(3y) = 25x^{2} + 9y^{2} + 30xy\)
- \(2^{2} = 4\)
- \(2(5x + 3y)(2) = 4(5x + 3y) = 20x + 12y\)

So we have:
\[ (5x + 3y + 2)^{2} = 25x^{2} + 9y^{2} + 30xy + 4 + 20x + 12y \]

### Step 3: Expand \((5x + 3y - 2)^{2}\)
Using the identity:
\[ (5x + 3y - 2)^{2} = (5x + 3y)^{2} + (-2)^{2} + 2(5x + 3y)(-2) \]
Calculating each part:
- \((5x + 3y)^{2} = 25x^{2} + 9y^{2} + 30xy\) (same as above)
- \((-2)^{2} = 4\)
- \(2(5x + 3y)(-2) = -4(5x + 3y) = -20x - 12y\)

So we have:
\[ (5x + 3y - 2)^{2} = 25x^{2} + 9y^{2} + 30xy + 4 - 20x - 12y \]

### Step 4: Combine both expansions
Now we add the two expanded forms:
\[ (25x^{2} + 9y^{2} + 30xy + 4 + 20x + 12y) + (25x^{2} + 9y^{2} + 30xy + 4 - 20x - 12y) \]

### Step 5: Simplify the combined expression
Combining like terms:
- \(25x^{2} + 25x^{2} = 50x^{2}\)
- \(9y^{2} + 9y^{2} = 18y^{2}\)
- \(30xy + 30xy = 60xy\)
- \(4 + 4 = 8\)
- \(20x - 20x = 0\)
- \(12y - 12y = 0\)

So, the simplified expression is:
\[ 50x^{2} + 18y^{2} + 60xy + 8 \]

### Final Answer
The final simplified expression is:
\[ 50x^{2} + 18y^{2} + 60xy + 8 \]

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