Simplify `(x+y+z)^(3)-(x-y-z)^(2)`.

Answer

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

### Step 1: Expand the expressions
First, we will expand both \((x+y+z)^{3}\) and \((x-y-z)^{2}\).

1. **Expand \((x+y+z)^{3}\)** using the binomial theorem:
\[ (x+y+z)^{3} = x^{3} + 3x^{2}(y+z) + 3x(y+z)^{2} + (y+z)^{3} \]
Expanding \((y+z)^{2}\) and \((y+z)^{3}\):
\[ (y+z)^{2} = y^{2} + 2yz + z^{2} \]
\[ (y+z)^{3} = y^{3} + 3y^{2}z + 3yz^{2} + z^{3} \]
So,
\[ (x+y+z)^{3} = x^{3} + 3x^{2}(y+z) + 3x(y^{2} + 2yz + z^{2}) + (y^{3} + 3y^{2}z + 3yz^{2} + z^{3}) \]

2. **Expand \((x-y-z)^{2}\)**:
\[ (x-y-z)^{2} = x^{2} - 2x(y+z) + (y+z)^{2} \]
Using \((y+z)^{2} = y^{2} + 2yz + z^{2}\):
\[ (x-y-z)^{2} = x^{2} - 2x(y+z) + (y^{2} + 2yz + z^{2}) \]

### Step 2: Combine the expansions
Now we have both expansions:
\[ (x+y+z)^{3} - (x-y-z)^{2} \]
Substituting the expansions:
\[ = \left(x^{3} + 3x^{2}(y+z) + 3x(y^{2} + 2yz + z^{2}) + (y^{3} + 3y^{2}z + 3yz^{2} + z^{3})\right) - \left(x^{2} - 2x(y+z) + (y^{2} + 2yz + z^{2})\right) \]

### Step 3: Simplify the expression
Now we will simplify this expression by combining like terms:
1. Combine \(x^{3}\) and \(x^{2}\):
\[ = x^{3} + (3x^{2}(y+z) - x^{2}) + 3x(y^{2} + 2yz + z^{2}) + (y^{3} + 3y^{2}z + 3yz^{2} + z^{3}) + 2x(y+z) - (y^{2} + 2yz + z^{2}) \]

2. Group the terms:
\[ = x^{3} + (3y + 3z - 1)x^{2} + (3y^{2} + 3z^{2} + 3yz + 2y + 2z - 1) \]

### Step 4: Final expression
The final simplified expression is:
\[ = 2x(2y + 2z) \]

### Final Answer:
\[ = 2x(2y + 2z) \]

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