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(a-b)^2(a+b)+ab(a+b)=a^3+b^3
Hello, Crystal09!
What's stopping you from multiply it out?Show that: .$\displaystyle (a-b)^2(a+b)+ab(a+b)\:=\:a^3+b^3$
If you looking for a deliberately complicated proof . . .
We have: .$\displaystyle a^3 + b^3$
Factor: .$\displaystyle (a + b)(a^2-ab+b^2)$
Subtract and add $\displaystyle ab\!:\;\;(a + b)(a^2 \:{\color{red}-\: ab}\: - ab + b^2 \:{\color{red}+\: ab}\,)$
And we have: .$\displaystyle (a + b)(a^2 - 2ab + b^2 + ab)$
Factor: .$\displaystyle (a+b)\left([a-b]^2 + ab\right)$
Distribute: .$\displaystyle (a+b)(a-b)^2 +(a+b)ab \;=\;(a-b)^2(a+b) + ab(a+b) $
Yeah, what's stopping you from multiply it out?
I can try to solve the half of the algebraic expression you're struggling with, in order to help you solving the remaining under;
$\displaystyle (a-b)^2 \cdot (a+b) + ab(a+b) = a^3 + b^3 $
$\displaystyle (a^2-2ab+b^2) \cdot (a+b) + (a^2b + ab^2) = a^3 + b^3 $