HMMT 1999 O10

$A, B, C, D$, and $E$ are relatively prime integers (i.e., have no single common factor) such that the polynomials $5 A x^{4}+4 B x^{3}+3 C x^{2}+2 D x+E$ and $10 A x^{3}+6 B x^{2}+3 C x+D$ together have 7 distinct integer roots. What are all possible values of $A$ ? Your team has been given a sealed envelope that contains a hint for this problem. If you open the envelope, the value of this problem decreases by 20 points. To get full credit, give the sealed envelope to the judge before presenting your solution.