A polynomial [tex]p(x)[/tex] has a root [tex]x_0[/tex] if it is a multiple of [tex](x-x_0)[/tex]
So, if you want roots [tex]-1, \sqrt{2}, \sqrt{3}[/tex], your polynomial must be a multiple of
[tex](x+1)\left(x-\sqrt{2}\right)\left(x-\sqrt{3}\right)[/tex]
If we add another monomial, we would raise the degree, but we want it to be as low as possible. So, we can only add a multiplicative constant: your polynomial is
[tex]p(x)=a(x+1)\left(x-\sqrt{2}\right)\left(x-\sqrt{3}\right)[/tex]
