(3,0),(-1,8): Quick answer.
Comment
There are several ways to do this. The first and most immediate way to to graph it. This graph came from Desmos. It shows you the immediate solution (of which there are two).
The graph tells us the answer immediately. The two intersect points that both graphs have in common are (3,0) and (-1,8)
Algebra
Of course on a test, you won't be able to that. You must do the problem algebraically. Notice that the ys have to be the same. That means that the right hand side of the two equations must be the same.
x^2 - 4x + 3 = -2x + 6 Make sure you know where this came from. I'm solving for x first and then I'll come back to do y
Transfer the right side to the left. Start by adding 2x to both sides.
x^2 - 4x + 2x + 3 = 6 Now subtract 6 from both sides.
x^2 - 2x + 3 - 6 = 0
x^2 - 2x - 3 = 0
This factors (you might have suspected that from the graph).
(x - 3)(x + 1) = 0 That means you have two x values.
x - 3 = 0
x = 3 (Didn't the graph predict that?)
x + 1 = 0
x = - 1 (And the graph predicted this as well).
Now go back to one of the original equations
y = - 2x + 6
If x = 3 then
y = -2*3 + 6
y = 0
First point of intersection = (3,0) <<<< answer
For x = -1
y = -2x + 6
y = -2*(-1) + 6
y = 2 + 6
y = 8
Second point of intersection equals (-1,8) <<<< answer