The greatest possible number of intersections between a quadratic expression and a linear expression is two.
Intersection of quadratic and linear expressions
A quadratic expression represents a parabola, and a linear expression represents a straight line. These two curves can intersect at most two times.
One intersection occurs when the parabola and the line touch each other at a single point. The second intersection occurs when the line crosses the parabola.
A scenario where is no intersections is possible if the parabola and the line do not intersect at any point or if they are parallel and never cross paths.
However, the question asks for the greatest possible number of intersections, so the answer is two.