smc

If $45$ is expressed as a product of five distinct integer factors, the absolute value of the product of any four is at least $|(-3)(-1)(1)(3)|=9$, so no factor can have an absolute value greater than $5$. Thus the factors of the given expression are five of the integers $\pm 3,\pm 1,\pm 5$. The product of all six of these is $-225=(-5)(45)$, so the factors are $-3,-1,1,3,$ and $5.$ The corresponding values of $a,b,c,d,$ and $e$ are $9,7,5,3,$ and $1,$ and their sum is $\boxed{25}$