Add :
`x^(3) - x^(2)y + 5xy^(2) + y^(3) , -x^(3) - 9xy^(2) + y^(3), 3x^(2)y + 9xy^(2)`

Answer

To solve the problem of adding the algebraic expressions \(x^{3} - x^{2}y + 5xy^{2} + y^{3}\), \(-x^{3} - 9xy^{2} + y^{3}\), and \(3x^{2}y + 9xy^{2}\), we will follow these steps:

### Step 1: Write down the expressions
We start by writing down the three expressions we need to add:
\[ x^{3} - x^{2}y + 5xy^{2} + y^{3} + (-x^{3} - 9xy^{2} + y^{3}) + (3x^{2}y + 9xy^{2}) \]

### Step 2: Remove the parentheses
Next, we remove the parentheses:
\[ x^{3} - x^{2}y + 5xy^{2} + y^{3} - x^{3} - 9xy^{2} + y^{3} + 3x^{2}y + 9xy^{2} \]

### Step 3: Combine like terms
Now we will group and combine the like terms:
- **Cubic terms**: \(x^{3} - x^{3} = 0\)
- **Quadratic terms**: \(-x^{2}y + 3x^{2}y = 2x^{2}y\)
- **Linear terms**: \(5xy^{2} - 9xy^{2} + 9xy^{2} = 5xy^{2}\)
- **Cubic terms in \(y\)**: \(y^{3} + y^{3} = 2y^{3}\)

So we have:
\[ 0 + 2x^{2}y + 5xy^{2} + 2y^{3} \]

### Step 4: Write the final expression
Thus, the final expression after combining like terms is:
\[ 2x^{2}y + 5xy^{2} + 2y^{3} \]

### Step 5: Factor out common terms (if necessary)
We can factor out \(y\) from the expression:
\[ y(2x^{2} + 5xy + 2y) \]

### Final Answer
The final answer is:
\[ y(2x^{2} + 5xy + 2y) \]

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