To find the points of intersection, substitute the expression for x from the line equation into the curve equation.

From x + 2y = 9, we have x = 9 - 2y.

Substitute into the curve equation xy + 18 = 0:

\((9 - 2y)y + 18 = 0\)

\(9y - 2y^2 + 18 = 0\)

\(-2y^2 + 9y + 18 = 0\)

Rearrange to form a quadratic equation:

\(2y^2 - 9y - 18 = 0\)

Solving this quadratic equation using the quadratic formula:

\(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

where \(a = 2\), \(b = -9\), \(c = -18\).

Calculate the discriminant:

\(b^2 - 4ac = (-9)^2 - 4 \times 2 \times (-18) = 81 + 144 = 225\)

Thus,

\(y = \frac{9 \pm 15}{4}\)

So, \(y = 6\) or \(y = -1.5\).

Substitute back to find \(x\):

For \(y = 6\):

\(x = 9 - 2(6) = -3\)

For \(y = -1.5\):

\(x = 9 - 2(-1.5) = 12\)

Thus, the coordinates of the points are (12, -1.5) and (-3, 6).