1. **State the problem:** We want to graph the quadratic function $$y = x^2 + 5x + 6$$. 2. **Formula and important rules:** A quadratic function is generally written as $$y = ax^2 + bx + c$$ where $a$, $b$, and $c$ are constants. The graph is a parabola. 3. **Find the roots (x-intercepts):** Solve $$x^2 + 5x + 6 = 0$$. 4. **Factor the quadratic:** $$x^2 + 5x + 6 = (x + 2)(x + 3)$$. 5. **Set each factor to zero:** $$x + 2 = 0 \Rightarrow x = -2$$ and $$x + 3 = 0 \Rightarrow x = -3$$. 6. **Vertex:** The vertex of a parabola $$y = ax^2 + bx + c$$ is at $$x = -\frac{b}{2a}$$. 7. **Calculate vertex x-coordinate:** $$x = -\frac{5}{2 \times 1} = -\frac{5}{2} = -2.5$$. 8. **Calculate vertex y-coordinate:** Substitute $$x = -2.5$$ into the function: $$y = (-2.5)^2 + 5(-2.5) + 6 = 6.25 - 12.5 + 6 = -0.25$$. 9. **Plot points:** The parabola passes through points $(-3,0)$, $(-2,0)$, and vertex $(-2.5, -0.25)$. 10. **Axis of symmetry:** The vertical line $$x = -2.5$$ is the axis of symmetry. 11. **Sketch the parabola:** It opens upwards since $a=1 > 0$. **Final answer:** The graph of $$y = x^2 + 5x + 6$$ is a parabola with roots at $x=-3$ and $x=-2$, vertex at $(-2.5, -0.25)$, and axis of symmetry at $x=-2.5$.