+130554 Here's one possibility........https://www.desmos.com/calculator/o9djnpu1cs
I know that roots with an even multiplicity will"kiss" the x axis........so.....allI had to do was construct the polynomial y = (x-3)^2(x+2)^2..and this polynomial will have "turning points" at (0,3) and (0,-2).
It will have a y intercept at (0,36), so all I had to do was add the -26 at the end, and this shifts the polynomial "down" by 26 units so that the "new" y intercept is (0,10). This does not affect the values of the turning points.
Of course, this polynomial has one more turning point when x =1/2, but, as far as I know, you didn't say it couldn't have more than two turning points...
