The red dots are the four dots from the question. These are calculated as follows:
The formula is already written in the form of #a \cdot x^2+b \cdot x +c# with #a =3#, #b=1# and #c=-2#. Seeing as #a>0# the graph is a parabola that opens upward.

The intersection with the #y#-axis is equal to the value of the constant in the quadratic formula, which is equal to #-2#. That means that the coordinates of the intersection point with the #y#-axis are #\rv{0,-2}#.

The #x#-value of the vertex is given by #x=-\dfrac{b}{2 \cdot a}# and is equal to:
\[\begin{array}{rclrl}
x&=& -\dfrac{1}{2 \cdot 3} &&\phantom{xxx}\blue{\text{formula entered}}\\
&=& -{{1}\over{6}} &&\phantom{xxx}\blue{\text{simplified}}\\
\end{array}\]
The #y#-value of the vertex is calculated by entering #x=-{{1}\over{6}}# in the formula. Which gives:
\[\begin{array}{rclrl}
y&=& 3 \cdot \left(-{{1}\over{6}}\right)^2 -{{1}\over{6}} -2
&&\phantom{xxx}\blue{\text{formula entered}}\\
&=& -{{25}\over{12}} &&\phantom{xxx}\blue{\text{calculated}}\\
\end{array}\]
The coordinates of the vertex are: #\rv{-{{1}\over{6}},-{{25}\over{12}}}#. To draw the point in the graph, we have to write the coordinates as decimal numbers (rounded to 1 decimal). That gives: #\rv{-0.2,-2.1}#.

The intersections with the #x#-axis are the points that correspond to #y=0#.
\[\begin{array}{rcl}
3\cdot x^2+x-2 &=& 0 \\&&\phantom{xxx}\blue{\text{the equation that should be calculated}}\\
x=\dfrac{-{1}-\sqrt{1^2-4 \cdot 3 \cdot -2}}{2 \cdot 3} &\vee& x=\dfrac{-{1}+\sqrt{1^2-4 \cdot 3 \cdot -2}}{2 \cdot 3} \\&&\phantom{xxx}\blue{\text{quadratic formula entered}}\\
x={{2}\over{3}} &\vee& x=-1 \\&&\phantom{xxx}\blue{\text{calculated}}\\
\end{array}\]
The coordinates of the intersections with the #x#-axis are: #\rv{{{2}\over{3}},0}# and #\rv{-1,0}#. To draw the point in the graph, we have to write the coordinates as decimal numbers (rounded to 1 decimal). That gives:#\rv{0.7,0}# and #\rv{-1,0}#.

The four points in the graph are: #\rv{0,-2}#, #\rv{-{{1}\over{6}},-{{25}\over{12}}}#, #\rv{{{2}\over{3}},0}# and #\rv{-1,0}#.