5x - 6 >= -4x^2
4x^2 + 5x - 6 >= 0
(4x - 3)(x + 2) >= 0
So either both brackets are positive, or both brackets are negative, which gives us
4x - 3 >= 0 and x + 2 >= 0 OR 4x - 3 <=0 and x + 2 <= 0
4x >= 3 and x >= -2 OR 4x <= 3 and x <= -2
x >= 3/4 and x >= -2 OR x <= 3/4 and x <= -2
the 1st pair of inequalities gives us: x >= 3/4 (both inequalities are satisfied by this)
the 2nd pair of inequalities gives us: x <= -2 (both inequalities are satisfied by this)
We have thus two regions of validity for the original inequality.
x <= -2, OR x > 3/4