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Basic math:
If you have an equation and divide both sides by the same factor, the equation remains valid, UNLESS if that factor is zero (remember: "thou shall never divide by zero?")
When you go from (a+b)(a-b) = b(a-b) => a+b = b
you are dividing both sides by (a-b); however, as a=b, (a-b)=0, you are dividing by zero, and the equation loses any validity
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