(i) Direct method.
$x^3+4 x=0$ or $x\left(x^2+4\right)=0$
$\Rightarrow x^2+4 \neq 0, \mathrm{x} \in \mathrm{R}$ hence, $x=0$
(ii) Method of contradiction.
Let $x \neq 0$ and let it be $x=p, \mathrm{p} \in \mathrm{R}, p$ is a root of $x^3+4 x=0$
$\therefore p^3+4 p=0 \quad \therefore p\left(p^2+4\right)=0$
$p \neq 0$ Also $\mathrm{p}^2+4 \neq 0 \Rightarrow p=0$
(iii) Contrapositive. $\mathrm{q}$ is not true
$\Rightarrow$ Let $x=0$ is not true
$\Rightarrow$ let $x=p \neq 0 \quad \therefore p^3+4 p=0$,
$p$ being the root of $x^2+4=0$
or $\mathrm{p}\left(\mathrm{p}^2+4\right)=0$, Now, $\mathrm{p}=0$
Also $\mathrm{p}^2+4 \neq 0 \Rightarrow p\left(p^2+4\right) \neq 0$ if $p$ is not true
$\therefore \quad x=0$ is the root of $x^3+4 x=0$