Basically find the minimum point. That is the lowest point of the curve so if that is above y=-3, the whole graph will be above it. Its a minimum point and not a maximum as the coefficient of x^2 is positive.
Left is completing the square right is differentiation.

Sorry for blur picture but just have to take two equations substitute together and equate to 0. using discriminant < 0 to find the range

For part 2, use discriminant then complete the square and simple explanation about the square is always greater than 0

Easier way to complete the square, you can google more on how to do it the quicker way