You have two points: x=-3, and x=0.
Test the areas around each point. I chose x=-4, x=-1, and x=1
So we know that before x=-3, the derivative of our function is less than zero, which means that our function is decreasing at this time.
Between x=-3 and x=0 the derivative becomes positive, so our function is increasing.
After x= 0 our function is still positive so our function is still increasing.
This means that our function is increasing at all points after x=-3... EXCEPT that we know the derivative equals zero at the point x=0, so at this point, it is neither increasing nor decreasing.
So f(x) is increasing on
You mistakenly set 4x^2 equal to 1, which is where you got 1/4, but you should have kept it equal to zero (because if it equals zero, then it is irrelevant what x+3 equals, since it will be multiplied by zero), and 0/4 = 0, and the square root of 0 = 0 so x=0