To solve the equation xy + x + y = 9, we can use a technique called factoring by grouping.

First, let's rearrange the equation to group the terms with xy and x together, and the terms with y separately:

xy + x + y = 9

(x + 1)y + x = 9

Now, let's factor out the common factor of (x + 1) from the terms with xy and x:

(x + 1)y + x = 9

(x + 1)(y + 1) = 9

Now, we have factored the left side of the equation. To solve for x and y, we need to find the values of (x + 1) and (y + 1) that multiply together to give 9.

The possible pairs of factors of 9 are (1, 9) and (-1, -9). Since we have (x + 1)(y + 1) = 9, we can set up two equations:

x + 1 = 1 and y + 1 = 9

or

x + 1 = -1 and y + 1 = -9

Solving the first equation, we get:

x + 1 = 1

x = 0

Substituting this value of x into the original equation, we get:

0y + 0 + y = 9

y = 9

So, one solution is x = 0 and y = 9.

Solving the second equation, we get:

x + 1 = -1

x = -2

Substituting this value of x into the original equation, we get:

-2y - 2 + y = 9

-y = 11

y = -11

So, another solution is x = -2 and y = -11.

Therefore, the solutions to the equation xy + x + y = 9 are (0, 9) and (-2, -11).