The x intercepts reveal the zeroes p(x)=a(x-1.5)(x-4)(x-10)(x-b).
We need to find a and b. If we multiply the factors we get a(x⁴-x³(15.5+b)+x²(61+15.5b)-x(60+61b)+60b).
If the y intercept is -3 then 60ab=-3, 20ab=-1.
p(x)≥0 in [4,10] so a=-1 and -20b=-1 and b=1/20.
p(x)=-(x⁴-15.55x³+61.775x²-63.05x+3) or:
-x⁴+15.55x³-61.775x²+63.05x-3.