First we will factorize:
x-9>0 and x+5>0
==> x>9 and x>-5
Then x belongs to the interval (9,infinity)....(1)
x-9<0 and x+5<0
x<9 and x< -5
Then x belongs to the interval (-infinity, -5)....(2)
From (1) and (2):
x belongs to the interval (-inf, -5)U(9,inf)
OR x = R-[-5, 9]
To solve x^2-4x-45 > 0.
The principle of solving the quadratic inequality like this( expressed as greater than zero) is finding the roots of the quadratic equation and see that the quadratic expression is positive only if the x values are outside the interval representented by the roots. This involves in factorising the quadratic expression and see that values of the variable x makes (i) both factors are positive or greater than zero, or (ii) both factors less than zero, so that their product is positive.
So x^2-4x-45 > 0. We factorise the left.
x^2-9x+5x-45 > 0
x(x-9) +5(x-9) > 0. Pull out (x-9).
(x-9)(x+5) > 0.
So for values of x < -5 makes both factors negative .So x< -5 makes the product positive.
Similarly for values of x > 9 both factors are positive. And so, the product of the facrors is positive.
Therefore, in the interval notation, when x belongs to (-infinity , -5) U (9 , infinity ) the given quadratic inequality holds good.