If the constant term in the binomial expansion of ( x - k x ^ 2 ) ^...

${T}_{r + 1} = Cr 10 {\left(\dfrac{- k}{{x}^{2}}\right)}^{r}{\left(\sqrt{x}\right)}^{10 - r}$

$= Cr \cdot 10 {\left(- k\right)}^{r} \cdot {x}^{5 - \dfrac{5r}{2}}$

for constant term

$⇒ 5 - \dfrac{5r}{2} = 0 ⇒ r = 2$

$⇒ {T}_{3 =} C2 10 \cdot {k}^{2} = 405$

$⇒ \dfrac{10 ( 9 )}{2}{k}^{2} = 405$

$⇒ {k}^{2} = 9$

$⇒ \left|k\right| = 3$