因为这句中有by using computers这个提示被动语态的部分.句子用了倒装,正确语序为：
They believed that the production of their factory would be increased greatly by using computers.
如果是主动的话,那个by using computers就显得很怪了
这句用主动的话应该是：
They believed that using computers would greatly increase the production of their factory.