Question:
Find the value
$a^{3}+3 a^{2} b+3 a b^{2}+b^{3}-8$
Solution:
$=(a+b)^{3}-8$
$\left[\therefore a^{3}+3 a^{2} b+3 a b^{2}+b^{3}=(a+b)^{3}\right]$
$=(a+b)^{3}-23$
$=(a+b-2)\left((a+b)^{2}+(a+b) \times 2+2^{2}\right)$
$=(a+b-2)\left(a^{2}+2 a b+b^{2}+2 a+2 b+4\right)$
$\therefore a^{3}+3 a^{2} b+3 a b^{2}+b^{3}-8=(a+b-2)\left(a^{2}+2 a b+b^{2}+2 a+2 b+4\right)$
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