It's hard to imagine making the algebra here any prettier, or more natural. And the result is very striking: a(ax^2 + 2bx + c) - (ax + b)^2 = ac - b^2 means that the polynomial analog of ac - b^2 turns out to have no x dependence -- and, indeed, is just ac - b^2 itself.
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My Twitter suggestion is to separate out those who are always attacking people from those who are willing to have a reasonable discussion. I've found that math people on Twitter are generally helpful, or at least have some nice perspective to offer.
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I've formed a similar impression. Although even among the mathematicians using Twitter I have noticed some who seem exceptionally sensitive to any slight, however unintentional, and who are prone to an unforgiving stance, no matter the apologies proffered. Nerve-wracking for me.
End of conversation
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