Hey, how would I start off solving

(x-5)(x+2)<0???

the easy way is to use what you know about the graphs of parabolas.

Since the coefficient of x^2 is positive, you know that the parabola open upward. Since the graph crosses the x-axis at the roots, the function is negative between the two roots, and positive below and above them.

The roots are at -2 and 5.
So, for -2 < x < 5 f(x) < 0

If you want to solve it algebraically, you know that if the product of two numbers is negative, exactly on of the two is negative. So, either

x-5 < 0 meaning x < 5
and x+2 > 0 meaning x > -2
OR
x-5 > 0 meaning x > 5
and x+2 < 0 meaning x < -2
But there is no number less than -2 and greater than 5. So, the first solution is the only one.

So, the product is negative if -2 < x < 5