If y'=2x+3 then,

y=x^2+3x+C, for some C ----> Anti-derivative

The problem says the curve (parabola in this case) intersects at (2,20) which means when x=2 then y=20:

20=4+6+C thus, C=10.

The unique curve must be,

y=x^2+3x+10

The line is,

y=3.6x+12.8

The problem is that these two curve intersect in one other point which is not (-3,10) as you say. Thus there is no such curve.