The middle term in the expansion of $(1+\alpha x )^{ n }$ is $\left(\frac{ n }{2}+1\right)^{\text {th }}$ term.

General term in the expansion of $(1+\alpha x )^4$ is $4_{ C _{ r }} a _{ r } x ^{ r }$

General term in the expansion of $(1-\alpha x)^6$ is $6_{C_1 r} a^r x^r$

The middle term in the expansion of $(1+\alpha x )^4$ is $\left(\frac{4}{2}+1\right)^{\text {th }}=3^{\text {rd }}$ term.

The middle term in the expansion of $(1-\alpha x )^6$ is $\left(\frac{6}{2}+1\right)^{\text {th }}=4^{\text {th }}$ term.

Given: $4_{ C _2 \alpha^2}=-6_{ C _3 \alpha^3}$

$\Rightarrow \alpha =\frac{-4_{ C 2}}{6_{ C 3}}$

$=\frac{-\frac{4 \times 3}{2 \times 1}}{\frac{6 \times 5 \times 4}{3 \times 2 \times 1}}$

$=\frac{-6}{20}=\frac{-3}{10}$