The number o f real negative terms in the binomial expansion of $(1+\mathrm{ix}{)}^{4\mathrm{n}-2},\mathrm{n}\in \mathrm{N},\mathrm{x}>0$ is

${\mathrm{T}}_{\mathrm{r}+1}{=}^{4\mathrm{n}-2}{\mathrm{C}}_{\mathrm{r}}(\mathrm{ix}{)}^{\mathrm{r}}$

${\mathrm{T}}_{\mathrm{r}+1}$ is negative, if i^r is negative and real.

$\begin{array}{r}{\mathrm{i}}^{\mathrm{r}}=-1\Rightarrow \mathrm{r}=2,6,10,\dots \text{ which form an A.P. }\\ \begin{array}{rl}0& \le \mathrm{r}\le 4\mathrm{n}-2\\ 4\mathrm{n}-2& =2+(\mathrm{r}-1)4\Rightarrow \mathrm{r}=\mathrm{n}\end{array}\end{array}$

The required number of terms is n.