To solve this system of equations, we will use the substitution method.

Given equations:
1) xy + x + y = 9
2) x^2y + xy^2 = 20

From equation 1, we can rewrite it as:
y(x + 1) + x = 9
y(x + 1) = 9 - x
y = (9 - x) / (x + 1)

Now, we substitute y into equation 2:
x^2((9 - x) / (x + 1)) + x((9 - x) / (x + 1))^2 = 20
Expanding and simplifying:
x(9 - x) + x(9 - x)^2 / (x + 1) = 20
9x - x^2 + x((81 - 18x + x^2) / (x + 1)) = 20
9x - x^2 + (81x - 18x^2 + x^3) / (x + 1) = 20
Multiplying by (x + 1) to clear the denominator:
9x(x + 1) - x^2(x + 1) + 81x - 18x^2 + x^3 = 20(x + 1)
9x^2 + 9x - x^2 - x + 81x - 18x^2 + x^3 = 20x + 20
Simplifying further:
8x^3 - x^2 + 10x = 20

This cubic equation can be solved to find the values of x and subsequently y.