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Đặt [TEX]a+b=m \ ; \ a-b=n[/TEX] thì [TEX]4ab=m^2-n^2[/TEX]
[TEX]a^3+b^3=(a+b)[(a-b)^2+ab]=m\(n^2+\frac{m^2-n^2}{4}\)[/TEX]
Ta có:
[TEX]A=(m+c)^3-4.\frac{m^3+3mn^2}{4}-4c^3-3c(m^2-n^2)[/TEX]
[TEX]A=m^3+3m^2c+3mc^2+c^3-m^3-3mn^2-4c^3-3cm^2+3cn^2[/TEX]
[TEX]A=3mc^2-3c^3-3mn^2+3cn^2[/TEX]
[TEX] A=3(mc^2-c^3-mn^2+cn^2)[/TEX]
[TEX] A=3[c^2(m-c)-n^2(m-c)][/TEX]
[TEX]\Rightarrow A=3(m-c)(c-n)(c+n)[/TEX]
Vậy [TEX]A=3(a+b-c)(c+a-b)(c-a+b)[/TEX]