You can put this solution on YOUR website! .
Show that there are no values of k for which the line y=3-2x can intersect the curve y=2x^2+kx+3
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The statement is WRONG.
The counter-example is k = 0 shown in the plot below.
Plots y = (red) and y = 3 - 2x (green)
What is the reason you post wrong problem to this forum ??