Simplify,` (x - y ) ^(2) + (x-y) - 6x (x ^(2) - y ^(2))`

Answer

To simplify the expression \( (x - y)^2 + (x - y) - 6x(x^2 - y^2) \), we will follow these steps:

### Step 1: Expand \( (x - y)^2 \)
Using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \):
\[ (x - y)^2 = x^2 - 2xy + y^2 \]

### Step 2: Rewrite the expression
Now, substituting the expanded form back into the expression:
\[ x^2 - 2xy + y^2 + (x - y) - 6x(x^2 - y^2) \]

### Step 3: Expand \( -6x(x^2 - y^2) \)
Recall that \( x^2 - y^2 = (x - y)(x + y) \):
\[ -6x(x^2 - y^2) = -6x(x - y)(x + y) \]
However, for simplification, we can directly expand it:
\[ -6x(x^2) + 6x(y^2) = -6x^3 + 6xy^2 \]

### Step 4: Combine all parts
Now, we combine all the parts:
\[ x^2 - 2xy + y^2 + x - y - 6x^3 + 6xy^2 \]

### Step 5: Rearrange the expression
Now, we can rearrange the terms:
\[ -6x^3 + x^2 + 6xy^2 - 2xy + y^2 + x - y \]

### Step 6: Write the final simplified expression
The final simplified expression is:
\[ -6x^3 + x^2 + 6xy^2 - 2xy + y^2 + x - y \]