(i) (m+n) $(m^{2} - m n + n^{2})$
(ii) (a-2b+5c) (a-b)-(a-b-c) (2a+3c)+(6a+b)(2c-3a-5b)

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Solution

i) $(m + n) (m^{2} - m n + n^{2})$
$= m^{3} - m^{2} n + m n^{2} + m^{2} n - m n^{2} + n^{3}$
$= m^{3} + n^{3}$
ii) $(a - 2 b + 5 c) (a - b) - (a - b - c) (2 a + 3 c) + (6 a + b) (2 c - 3 a - 5 b)$
$= a^{2} - a b - 2 a b + 2 b^{2} + 5 a c - 5 b c - 2 a^{2} - 3 a c + 2 a b + 3 b c + 2 a c + 3 c^{2}$ $+ 12 a c - 18 a^{2} - 30 a b + 2 b c - 3 a b - 5 b^{2}$
$= a^{2} - 19 a^{2} - 3 b^{2} + 3 c^{2} - 34 a b + 16 a c$

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∣ ∣ ∣ ∣ \begin{matrix} a & a + b & a + b + c 2 a & 3 a + 2 b & 4 a + 3 b + 2 c 3 a & 6 a + 3 b & 10 a + 6 b + 3 c \end{matrix} ∣ ∣ ∣ ∣ = a^{3}

∣ ∣ ∣ ∣ \begin{matrix} a & a + b & a + b + c 2 a & 3 a + 2 b & 4 a + 3 b + 2 c 3 a & 6 a + 3 b & 10 a + 6 b + 3 c \end{matrix} ∣ ∣ ∣ ∣

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