Recall that the general form of a parabola is as follows
The parabola is symmetric with respect the the showned vertical line, which is a line that passes through the vertex of the parabola. By this, we mean that if two points a,b happen to have f(a) = f(b), this means that the point a is at the same distance from the vertex as b.
Our function is f(x)= x^2+5x-3
Note that f(-3) = (-3)^2+5(-3)-3 = 9-15-3 = -9
Also, note that f(-2) = (-2)^2+5(-2)-3 = 4-10-3 = -9
Since f(-3) = f(-2) = -9, this means that -3 is at the same distance from the vertex as -2. This also means that the vertex x-coordinate lies in the middle between -3 and -2.
Note that the distance from -3 to -2 is 1 unit. So the vertex is at 1/2 distance from each number. So to find the x coordinate of the vertex, we simply add 1/2 to -3. This leads to
So the x coordinate of the vertex is -5/2.