make up a polynomial that has the following characteristics: crosses the x axis at -1 and 4 , touches the a-axis at 0 and 2 and is above the a axis between 0 and 2.
What do you mean by the a-axis? If this is just a typo (and a recurring one at that), then
$\displaystyle p(x)={\color{green}c}\cdot {\color{red}(x+1)(x-4)}{\color{blue}x^2(x-2)^2}$
will do. In order to have $\displaystyle p(x) > 0$ for $\displaystyle x\in ]0;2[$ you need to make $\displaystyle c<0$.