SOLUTION: The question states: Find the descriminant and the nature of the roots for 4x^2+2x+5=0. I know the descriminant is -76 but what is the nature of the roots?

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If the discriminant is positve, the solution has two real roots.
If the discriminant is negative, the solution has two complex conjugate roots.
If the discriminant is zero, the solution has one real root. This is also known as a double root.
In your problem, since the discriminant is negative (-76), the nature of the roots is they are complex conjugate roots.
If your quadratic equation were to be graphed, you would see that the curve (a parabola) does not intersect the x-axis.
The roots of the quadratic equation are simply the points on the x-axis where the curve intersects the x-axis.
Here's the graph of your equation so you can see the meaning of this.