Using the factor method, simplify `(x^(2)y+2xy+x+2)div(xy+1)`.

Answer

To simplify the expression \((x^2y + 2xy + x + 2) \div (xy + 1)\) using the factor method, we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{x^2y + 2xy + x + 2}{xy + 1} \]

### Step 2: Factor the numerator
We will factor the numerator \(x^2y + 2xy + x + 2\). We can group the terms:
\[ (x^2y + 2xy) + (x + 2) \]
Now, we can factor out the common terms in each group:
- From the first group \(x^2y + 2xy\), we can factor out \(xy\):
\[ xy(x + 2) \]
- From the second group \(x + 2\), we can factor out \(1\):
\[ 1(x + 2) \]

Now we can combine these:
\[ xy(x + 2) + 1(x + 2) = (xy + 1)(x + 2) \]

### Step 3: Substitute back into the expression
Now we substitute the factored form back into the expression:
\[ \frac{(xy + 1)(x + 2)}{xy + 1} \]

### Step 4: Cancel the common factors
Since \(xy + 1\) is present in both the numerator and the denominator, we can cancel it out:
\[ x + 2 \]

### Final Answer
Thus, the simplified expression is:
\[ \boxed{x + 2} \]

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