You need to decompose the polynomials on the LHS into a product of factors which when multiplied give the original form.
It should be clear that for and . It can easily be checked that for values between -3/4 and 2 it is negative.
See also http://www.wolframalpha.com/input/?i=4+x^2+-+5+x+-+6+<%3D+0
The examples, like "x2 + 2x – 8 < 0" and "–x2 + 6x – 9 > 0" are quadratic, they have "zero" on the other side of the inequality, and they involve factoring, just as you've been advised to proceed here. How are those examples "truly" different from what you are doing?