Factorise by grouping method :
`x^(2) + y^(2) + x + y + 2xy`

Answer

To factorise the expression \( x^2 + y^2 + x + y + 2xy \) by grouping method, follow these steps:

### Step 1: Rearrange the expression
We can rearrange the given expression as follows:
\[ x^2 + y^2 + 2xy + x + y \]

### Step 2: Identify a perfect square
Notice that the first three terms \( x^2 + y^2 + 2xy \) can be recognized as a perfect square. Recall the formula:
\[ (a + b)^2 = a^2 + b^2 + 2ab \]
Here, we can let \( a = x \) and \( b = y \). Thus, we can rewrite:
\[ x^2 + y^2 + 2xy = (x + y)^2 \]

### Step 3: Substitute back into the expression
Now we can substitute this back into our rearranged expression:
\[ (x + y)^2 + x + y \]

### Step 4: Factor out the common term
Notice that both terms \( (x + y)^2 \) and \( x + y \) have a common factor of \( (x + y) \). We can factor this out:
\[ (x + y)((x + y) + 1) \]

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