You understood the problem incorrectly.You proved the following statement.
For every t there exists an x such that for every a > x we have at^2 + 2t + 12 > 0.
Here x may depend on t (think about "every person has a farther": the father depends on the person).So you found an x = (-12 - 2t) / t^2 that depends on t.
You were supposed to prove the following.
There exists an x such that for every a > x and for every t we have at^2 + 2t + 12 > 0.
Here x may not depend on anything; it must be a concrete number. You need to find (the minimal) such number and prove that for every a > x and for every t we have at^2 + 2t + 12 > 0.
Note that the graph of at^2 + 2t + 12 is a parabola whose branches point up if a > 0 and down if a < 0. In order for the whole graph to be above the x-axis, we must have a > 0 and the equation at^2 + 2t + 12 must have no real roots.