The expression
`(x-y) (x^2+xy+y^2)+(x+y)(x^2-xy+y^2)-(x+y)(x^2-y^2)` is equal to

Options

A. `x^3-y^3+xy(x+y)`
B. `y^3-x^3+xy(y+x)`
C. `x^3+y^3+xy(y-x)`
D. `x^3+y^3+xy(x-y)`

Answer

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

### Step 1: Expand Each Term
We will expand each of the three terms in the expression.

1. **Expand** \((x-y)(x^2+xy+y^2)\):
\[ = x(x^2 + xy + y^2) - y(x^2 + xy + y^2) = x^3 + x^2y + xy^2 - (yx^2 + y^2x + y^3) = x^3 + x^2y + xy^2 - yx^2 - y^2x - y^3 = x^3 - y^3 \]

2. **Expand** \((x+y)(x^2-xy+y^2)\):
\[ = x(x^2 - xy + y^2) + y(x^2 - xy + y^2) = x^3 - x^2y + xy^2 + yx^2 - y^2x + y^3 = x^3 + y^3 \]

3. **Expand** \(-(x+y)(x^2-y^2)\):
\[ = -(x+y)(x-y)(x+y) = -(x^3 - y^3) \]

### Step 2: Combine the Expanded Terms
Now we will combine all the expanded terms:
\[ (x^3 - y^3) + (x^3 + y^3) - (x^3 - y^3) \]

### Step 3: Simplify the Expression
Now we will simplify the combined expression:
\[ = x^3 - y^3 + x^3 + y^3 - x^3 + y^3 \]
\[ = x^3 + y^3 - y^3 \]
\[ = x^3 \]

### Final Result
Thus, the expression simplifies to:
\[ \boxed{x^3} \]