Simplify: $(3 k - 4 r - 2 m)^{2} - (3 k + 4 r - 2 m)^{2}$

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Solution

$= (3 k - 4 r - 2 m)^{2} - (3 k + 4 r - 2 m)^{2}$
$= [(3 k)^{2} + (- 4 r)^{2} + (- 2 m)^{2} + {2 \times (3 k) \times (- 4 r)}$
$+ {2 \times (- 4 r) \times (- 2 m} + {2 \times (3 k) \times (- 2 m)} -]$
$- [(3 k)^{2} + (4 r)^{2} + (- 2 m)^{2} + {2 \times (3 k) \times (4 r)}$
$+ {2 \times 4 r \times (- 2 m)} + {2 \times (3 k) \times (- 2 m)}]$
By using the formula,
$(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 a c$
$= [9 k^{2} + 16 r^{2} + 4 m^{2} + {- 24 k r} + {16 r m} + {- 12 k m}]$
$- [9 k^{2} + 16 r^{2} + 4 m^{2} + {24 k r} + {- 16 r m} + {- 12 k m}]$
$= [9 k^{2} + 16 r^{2} + 4 m^{2} - 24 k r + 16 r m - 12 k m]$
$- [9 k^{2} + 16 r^{2} + 4 m^{2} + 24 k r - 16 r m - 12 k m]$
$= 9 k^{2} + 16 r^{2} + 4 m^{2} - 24 k r + 16 r m - 12 k m - 9 k^{2} - 16 r^{2}$ $- 4 m^{2} - 24 k r + 16 r m + 12 k m$
$= - 48 k r + 32 r m$
$= 32 r m - 48 k r$
$∴ (3 k - 4 r - 2 m)^{2} - (3 k + 4 r - 2 m)^{2} = 32 r m - 48 k r$

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