1. **State the problem:** We are given the quadratic function $y = x^2 + 5x + 6$ and want to analyze it. 2. **Formula and rules:** A quadratic function is generally written as $y = ax^2 + bx + c$ where $a$, $b$, and $c$ are constants. Here, $a=1$, $b=5$, and $c=6$. 3. **Find the roots (x-intercepts):** To find the roots, solve $x^2 + 5x + 6 = 0$. 4. **Factor the quadratic:** $x^2 + 5x + 6 = (x + 2)(x + 3)$. 5. **Set each factor to zero:** - $x + 2 = 0 \Rightarrow x = -2$ - $x + 3 = 0 \Rightarrow x = -3$ 6. **Find the vertex (extremum):** The vertex $x$-coordinate is given by $x = -\frac{b}{2a} = -\frac{5}{2 \times 1} = -\frac{5}{2} = -2.5$. 7. **Calculate the vertex y-coordinate:** Substitute $x = -2.5$ into the function: $$y = (-2.5)^2 + 5(-2.5) + 6 = 6.25 - 12.5 + 6 = -0.25$$ 8. **Interpretation:** The parabola opens upwards (since $a=1 > 0$), has roots at $x = -3$ and $x = -2$, and a minimum point (vertex) at $(-2.5, -0.25)$. **Final answer:** The roots are $x = -3$ and $x = -2$, and the vertex is at $(-2.5, -0.25)$.