Verify if the curves y=x^2+x+1 and y=-x^2-2x+6 have common points?

We'll equate the given
expressions:

x^2+x+1
=-x^2-2x+6

We'll shift all terms to one
side:

x^2 + x^2 + x + 2x + 1 - 6 =
0

We'll combine like
terms:

2x^2 + 3x - 5 = 0

We'll
apply quadratic fromula:

x1 =
[-3+sqrt(9+40)]/4

x1 =
(-3+7)/4

x1 = 1

x2 =
(-3-7)/4

x2 = -10/4

x2 =
-5/2

We'll get y1 coordinate, when x1 =
1:

y1 = 1+1+1=3 (we notice that we'll get the same value if
we'll replace x by 1 in the 2nd equation).

We'll get y2
coordinate, when x2 = -5/2:

y2 = 25/4 - 5/2 +
1

y2 = 15/4 + 1

y2 =
19/4

The intercepting points of the given
curves are: (1 ; 3) and (-5/2 ; 19/4).