To find out if there is point where the graphs intersect simply plug in one equation for another,
y = 4x + 3
Y = -x^2 - 1.
4x + 3 = -x^2 - 1
Then solve
4x + 3 = -x^2 - 1
X^2 + 4x + 3 = -1
X^2 + 4x + 4 = 0.
(X+2)(x+2) = 0.
There is one solution X = -2. To solve for the y coordinate plug X = -2 to any of the above equations, easier for the first one.
Y = 4x + 3
Y = 4(-2) + 3
Y = -8 + 3
Y = -5.
Point - P(-2,-5). Plug these points in both original equations and if in each the left side is equal to the right, then that is the point of intersection. Basically the graph intersects once.