The coefficient of the middle term in the binomial expansion in powers of $x$ of $(1+\alpha x{)}^4$ and of $(1-\alpha x{)}^6$ is the same if $\alpha$ equals

In the expansion of $(1+\alpha x{)}^4,$ the number of terms is $5.$ The middle term is the $3$ rd term ($T_3$).
$T_3={}^4{C}_2(\alpha x{)}^2=6{\alpha }^2{x}^2$.
Coefficient of middle term $=6{\alpha }^2.$ In the expansion of $(1-\alpha x{)}^6,$ the number of terms is $7.$ The middle term is the $4$ th term ($T_4$).
$T_4={}^6{C}_3(-\alpha x{)}^3=20(-{\alpha }^3)x^3=-20{\alpha }^3{x}^3$.
Coefficient of middle term $=-20{\alpha }^3.$ Given the coefficients are same:$6{\alpha }^2=-20{\alpha }^3$.
$6=-20\alpha$ (assuming $\alpha \ne 0$)
$\alpha =-\frac{6}{20}=-\frac{3}{10}$.