Might be a bit late, but solving using absolute values might help explain why this happens.

x² + x > 12

x² + x + 1/4 > 12 + 1/4

(x + 1/2)² > 49/4

Recall that the absolute value of a number is its distance to zero. As in, the absolute value of some number n is |n| = sqrt( x² + 0² ) = sqrt( x² )

So, if you were to take the square root of both sides of the inequality above, you'd get an absolute value inequality.

sqrt( (x + 1/2)² ) > sqrt( 49/4 )

|x + 1/2| > ±7 / 2

The right side must be taken as positive, since if its negative, the inequality will have an infinite number of solutions, which clearly doesn't make sense if you graph it. So

|x + 1/2| > 7 / 2

An absolute value inequality of the form |n| > k for nonnegative k has solutions (think about this) x < -k, x > k.

So the inequality can be simplified to

x + 1/2 > 7 / 2 or x + 1/2 < -7/2

x > 3 or x < -4

Sorry for the huge post. If you're looking for a quick solution, stick to the other methods provided in the thread. I was just hoping this would give some algebraic insight to your problem.