To be clearer, do not let the relation to 0 fool you. The relation can be with a term or a value other than zero on one side. Merely adding the additive inverse to both sides can bring the inequality to relate a quadratic expression to zero. Also, if the quadratic is factorable, you may be able to have something like (ax+b)(cx+d) (relation-symbol)(0).
Lym Y K,
The Mathispower4u (which jedishru posted) video you should find very helpful in understanding what to do with solving a quadratic inequality. The roots of the quadratic expression form the x-number line into three intervals, and any value in each interval can be chosen to test the truth or falsity for the interval.
The example in the paper in your included photograph shows [itex]-4x^2-4x-1[/itex] and we must assume it's meant as related versus 0. You can/should DIVIDE both sides by NEGATIVE 4, and this MUST reverse the direction of the inequality symbol. Why? because multiplication or division by a negative VALUE.
That step now gives you [itex]x^2+x+1/4[/itex] versus 0, as said, with relation reversed from what it was originally. This quadratic is factorable giving you exactly ONE critical x value.
[itex](x+1/2)^2[/itex]. versus 0. (You did not show on your paper the inequality symbol relating). The critical value is at [itex]x=-1/2[/itex], just one single value, cutting the x-number line into just two intervals. Now, you test each interval, and maybe also you need to test that critical x value of [itex]-1/2[/itex].