Factorise:
`y ^(2) - xy (1-x)-x ^(3)`

Answer

To factorise the expression \( y^2 - xy(1-x) - x^3 \), we can follow these steps:

### Step 1: Expand the expression
First, we need to expand the term \( -xy(1-x) \):
\[ y^2 - xy(1-x) - x^3 = y^2 - xy + x^2y - x^3 \]

### Step 2: Rearrange the terms
Now, let's rearrange the terms:
\[ y^2 + x^2y - xy - x^3 \]

### Step 3: Group the terms
Next, we can group the terms to factor them more easily:
\[ (y^2 + x^2y) + (-xy - x^3) \]

### Step 4: Factor out the common factors
Now, we can factor out the common factors from each group:
1. From \( y^2 + x^2y \), we can factor out \( y \):
\[ y(y + x^2) \]
2. From \( -xy - x^3 \), we can factor out \( -x \):
\[ -x(y + x^2) \]

Putting it all together, we have:
\[ y(y + x^2) - x(y + x^2) \]

### Step 5: Factor out the common binomial
Now, we can see that \( (y + x^2) \) is a common factor:
\[ (y + x^2)(y - x) \]

### Final Answer
Thus, the factorised form of the expression \( y^2 - xy(1-x) - x^3 \) is:
\[ (y + x^2)(y - x) \]

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