## Factoring (xy+1)(x+1)(y+1)+xy

This article will guide you through the process of factoring the expression **(xy+1)(x+1)(y+1)+xy**.

### Expanding the Expression

First, we need to expand the expression by multiplying out the brackets.

Let's start by multiplying the first two brackets:

(xy+1)(x+1) = xy(x+1) + 1(x+1) = x²y + xy + x + 1

Now, let's multiply this result by (y+1):

(x²y + xy + x + 1)(y+1) = x²y(y+1) + xy(y+1) + x(y+1) + 1(y+1) = x²y² + x²y + xy² + xy + xy + x + y + 1

Simplifying this, we get:

**x²y² + x²y + xy² + 2xy + x + y + 1**

Finally, let's add the term **xy** back in:

**x²y² + x²y + xy² + 2xy + x + y + 1 + xy**

This gives us the expanded form of our original expression.

### Factoring by Grouping

We can now factor this expression by grouping terms. Notice that we can group the first four terms together and the last three terms together:

(x²y² + x²y + xy² + 2xy) + (x + y + 1)

Now, let's factor out common factors from each group:

xy(xy + x + y + 2) + (x + y + 1)

We can see that the expression in the first group is very close to the expression in the second group. To make them identical, we can factor out a **1** from the second group:

xy(xy + x + y + 2) + 1(x + y + 1)

Now we have two identical expressions in the brackets:

(xy + 1)(xy + x + y + 2)

### Final Factoring

The last step is to factor the second expression in the brackets by grouping. We can group the first two terms and the last two terms:

(xy + 1)(xy + x + y + 2) = (xy + 1)(x(y + 1) + (y + 1))

Finally, we can factor out the common factor (y+1):

**(xy + 1)(y+1)(x+1)**

### Conclusion

We have successfully factored the expression **(xy+1)(x+1)(y+1)+xy** into **(xy + 1)(y+1)(x+1)**. This process involves expanding the expression, grouping terms, factoring out common factors, and simplifying.