What factors add to get - 18 and multiply to get 75?

There are no whole-number solutions. Assuming you are trying to factorise the quadratic expression x^2 - 18x + 75, the discriminant (b^2 - 4ac) is 324 - 300 = 24. Although it is positive, so there are two solutions, 24 is not a square number so it does not factorise cleanly.

To find the roots of the equation, I would complete the square:

(x - 9) ^2 - 81 + 75 = 0

(x - 9) ^2 - 6 = 0

(x - 9) ^2 = 6

x - 9 = ± √6

x = 9 ±√6

x = (9 + √6) or (9 - √6)