To solve this, we are looking for a graph that represents the system of equations given:

1. $y = x^2 + 2x + 3$, which is a parabola opening upwards.
2. $2y - 2x = 10$, which simplifies to $y = x + 5$, a straight line with slope 1 and y-intercept 5.

The solution to the system will occur where these two equations intersect. We need to identify a graph that shows both:

• A parabola that matches $y = x^2 + 2x + 3$,
• A straight line with the characteristics $y = x + 5$.

Key Observations:

• The parabola's vertex is at $x = -1$, $y = 2$, and it opens upwards.
• The line $y = x + 5$ has a slope of 1 and cuts the y-axis at $y = 5$.

Evaluation of Graphs:

From the given image:

• Graph C shows both the correct parabola and the correct straight line.

Thus, Graph C is the correct answer.

Would you like a detailed algebraic or graphical explanation of this?

Here are 5 follow-up questions:

1. How do you algebraically find the points of intersection for a system of equations?
2. What is the general form of a parabola and how does its shape depend on its coefficients?
3. How can you check if a straight line is tangent to a parabola?
4. What happens if a parabola and line have no points of intersection?
5. How does graphing help in verifying solutions to systems of equations?

Tip: To solve these systems algebraically, substitute the linear equation into the quadratic equation to find the intersection points.