To solve the problem, we need to determine the points of intersection between the curves described by the equations:

1. $y = x + 6$
2. $y = -x^2 + 5x + 6$

Step 1: Set the Equations Equal to Each Other

We find the intersection points by setting the equations equal to each other:

$x + 6 = -x^2 + 5x + 6$

Step 2: Simplify and Rearrange the Equation

Subtract $x + 6$ from both sides to get:

$0 = -x^2 + 4x$

Factor the equation:

$0 = x(-x + 4)$

Step 3: Solve for $x$

The solutions are:

$x = 0 \quad \text{or} \quad x = 4$

Step 4: Calculate the