$(i)$ $x^{2}+x$

$\because $  The degree of $x ^{2}+ x$ is  $2$  .                  $\therefore $ It is a quadratic polynomial.

$(ii)$ $x-x^{3}$

$\because$ The degree of $x-x^{3}$ is $3$.                        $\therefore$ It is a cubic polynomial.

$(iii)$ $y+y^{2}+4$

$\because$ The degree of $y+y^{2}+4$ is $2$                  $\therefore$ It is a quadratic polynomial.