1. State the problem. Problem: Find the domain and range of the function $f(x)=x^2+6x+5$. 2. Formula and important rules. The domain of any quadratic function is all real numbers so domain is $\mathbb{R}$. The x-coordinate of the vertex is given by $x_v=-\frac{b}{2a}$ and the y-coordinate is $y_v=f(x_v)$. If $a>0$ the parabola opens upward and the range is $[y_v,\infty)$. If $a<0$ the parabola opens downward and the range is $(-\infty,y_v]$. 3. Intermediate work and evaluation. Identify coefficients: $a=1$, $b=6$, $c=5$. Compute the vertex x-coordinate: $x_v=-\frac{b}{2a}=-\frac{6}{2\cdot 1}=-3$. Evaluate the function at the vertex: $f(-3)=(-3)^2+6(-3)+5=9-18+5=-4$. 4. Conclusion and final answer. Domain: $\mathbb{R}$. Range: $[-4,\infty)$. Final answer: Domain $\mathbb{R}$, Range $[-4,\infty)$.