We have to establish the quadrant where the vertex of the parable is located.

We know that the coordinates of the parabola vertex are:

V(-b/2a;-delta/4a), where a,b,c are the coefficients of the function and delta=b^2 -4*a*c.

y=f(x)=x^2 + 5x + 1

We'll identify the coefficients:

a=1, b=5, c=1, 2a=2, 4a=4

delta=5^2 -4*1*1=25-4=21

V(-b/2a;-delta/4a)=V(-5/2;-21/4)

Because the coordinates are both negative, the vertex is located in the third quadrant: V(-5/2;-21/4).

f(x) = x^2+5x+1.

To find the vertex.

y = x^+5x+1 =

=(x-5/2)^2 - (5/2)^2+1

= (x-5/2)^2 - (25-4)/4 =

=(x-5/2)^2- 21/4

y+21/4 = (x-5/2)^2 which is ciparable with the standard parabola y = X^2 whose vertex is at (0,0).

So the parabola y-21/4 = (x-5/2)^2 has the vertex V at x =5/2 and y = -21/4. Or V(x,y) = (5/2 , -21/4).