What are the coordinates of the point(s) of intersection for the equations y = -2x + 1 and y = x^2 + x + 3?

To find the coordinates of the points of intersection between the two equations, you need to set them equal to each other since at the intersection points, both equations will yield the same value for (y).

You have the linear equation (y = -2x + 1) and the quadratic equation (y = x^2 + x + 3). Setting them equal gives:

[

-2x + 1 = x^2 + x + 3

]

Rearranging this equation leads to:

[

0 = x^2 + 3x + 2

]

This can be factored into:

[

0 = (x + 1)(x + 2)

]

Setting each factor to zero gives the solutions:

[

x + 1 = 0 \Rightarrow x = -1

]

[

x + 2 = 0 \Rightarrow x = -2

]

Now, substituting these (x) values back into the linear equation to find the corresponding (y) values:

For (x = -1):

[

y = -2(-1) + 1 = 2 + 1 = 3