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What should be added to $$4c(-a - 2b + c)$$ to obtain $$3c(a - b + c) - 2c(a + b - c)$$?
The target is $$3ac - 3bc + 3c^2 - 2ac - 2bc + 2c^2 = ac - 5bc + 5c^2$$, and the given expression is $$-4ac - 8bc + 4c^2$$. The required expression is their difference, $$(ac - 5bc + 5c^2) - (-4ac - 8bc + 4c^2) = 5ac + 3bc + c^2$$.
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