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 =5#, #b=2# and #c=-4#. 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 #-4#. That means that the coordinates of the intersection point with the #y#-axis are #\rv{0,-4}#.

The #x#-value of the vertex is given by #x=-\dfrac{b}{2 \cdot a}# and is equal to:
\[\begin{array}{rclrl}
x&=& -\dfrac{2}{2 \cdot 5} &&\phantom{xxx}\blue{\text{formula entered}}\\
&=& -{{1}\over{5}} &&\phantom{xxx}\blue{\text{simplified}}\\
\end{array}\]
The #y#-value of the vertex is calculated by entering #x=-{{1}\over{5}}# in the formula. Which gives:
\[\begin{array}{rclrl}
y&=& 5 \cdot \left(-{{1}\over{5}}\right)^2 +2 \cdot -{{1}\over{5}} -4
&&\phantom{xxx}\blue{\text{formula entered}}\\
&=& -{{21}\over{5}} &&\phantom{xxx}\blue{\text{calculated}}\\
\end{array}\]
The coordinates of the vertex are: #\rv{-{{1}\over{5}},-{{21}\over{5}}}#. 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,-4.2}#.

The intersections with the #x#-axis are the points that correspond to #y=0#.
\[\begin{array}{rcl}
5\cdot x^2+2\cdot x-4 &=& 0 \\&&\phantom{xxx}\blue{\text{the equation that should be calculated}}\\
x=\dfrac{-{2}-\sqrt{2^2-4 \cdot 5 \cdot -4}}{2 \cdot 5} &\vee& x=\dfrac{-{2}+\sqrt{2^2-4 \cdot 5 \cdot -4}}{2 \cdot 5} \\&&\phantom{xxx}\blue{\text{quadratic formula entered}}\\
x={{-\sqrt{21}-1}\over{5}} &\vee& x={{\sqrt{21}-1}\over{5}} \\&&\phantom{xxx}\blue{\text{calculated}}\\
\end{array}\]
The coordinates of the intersections with the #x#-axis are: #\rv{{{-\sqrt{21}-1}\over{5}},0}# and #\rv{{{\sqrt{21}-1}\over{5}},0}#. To draw the point in the graph, we have to write the coordinates as decimal numbers (rounded to 1 decimal). That gives:#\rv{-1.1,0}# and #\rv{0.7,0}#.

The four points in the graph are: #\rv{0,-4}#, #\rv{-{{1}\over{5}},-{{21}\over{5}}}#, #\rv{{{-\sqrt{21}-1}\over{5}},0}# and #\rv{{{\sqrt{21}-1}\over{5}},0}#.