OR, you can forgo "Scapegoat" in the above combo and use your opponent's monsters instead, if you use "Super Polymerization".
OR, you could use "D. Tribe" and/or "DNA Surgery" (declaring Dragon-Type) with "Scapegoat" and "Fusion Gate" "Field Spell Card" on the field, in order to Summon "Five-Headed Dragon" so effectively and effortlessly indeed.
If you have "Lord of D." or "King Dragun" on the field, "Five-Headed Dragon" can't be targeted by Spell, Trap or monster effects.
You can use "DNA Surgery" along with "Super Polymerization" to Summon "Five-Headed Dragon". Use "DNA Surgery" to declare all monsters on the field to Dragon-Type, then use "Super Polymerization" to either on your opponent's or your own monsters to bring out "Five-Headed Dragon". It is also very effective to use your opponent's monsters as Fusion Materials, so that way, they won't have any monsters to defend from a direct attack.
"Painful Choice" can easily deposit Dragons into the Graveyard for the usage of "Dragon's Mirror".
This is a great card to use as a Tribute for "The Tyrant Neptune", as it can gain this card's ATK and effect. Tributing "Hardened Armed Dragon" in addition to this combo will grant "The Tyrant Neptune" additional protection.
"Dimension Fusion" can be used to Summon the Fusion Materials that were banished by "Dragon's Mirror".