jmc

Since $AB\parallel DE,$ we know that $\angle A=\angle E$ and $\angle B=\angle D.$ That works out nicely, since that means $△ABC\sim EDC.$ If $BD=4BC,$ that means $CD=BD-BC=3BC.$ Therefore, the ratio of sides in $ABC$ to $EDC$ is $1:3,$ meaning the ratio of their areas is $1:9.$

Since the area of $△ABC$ is $6{\text{ cm}}^2,$ that means the area of $△CDE$ is $\boxed{54}{\text{ cm}}^2.$