Answer
(a) See graph (red curve).
(b) domain $(-\infty,\infty)$, range $[\frac{14}{3},\infty)$.
(c) decreasing on $(-\infty,-\frac{1}{3})$, increasing on $(-\frac{1}{3},\infty)$.
Work Step by Step
(a) To graph $y=3x^2+2x+5=3(x+\frac{1}{3})^2+\frac{14}{3}$, start from $y=x^2$, shift the curve $\frac{1}{3}$ unit(s) to the left, stretch vertically by a factor of 3, then shift $\frac{14}{3}$ unit(s) up. See graph (red curve).
(b) We can determine the domain $(-\infty,\infty)$, range $[\frac{14}{3},\infty)$.
(c) The function is decreasing on $(-\infty,-\frac{1}{3})$, increasing on $(-\frac{1}{3},\infty)$.
