Find the domain and range of $LaTeX: \displaystyle f(x)=3 x^{2} + 60 x + 312$

The domain of every polynomial is all real numbers. The sign of $LaTeX: \displaystyle a$ is positive so the function has a minimum value at the vertex. The vertex is located at $LaTeX: \displaystyle h=\frac{-b}{2a}=-10$ and $LaTeX: \displaystyle k=f(h)=12$ . This gives the range as $LaTeX: \displaystyle [12,\infty)$