Simplify: $(a + b) (c - d) + (a - b) (c + d) + 2 (a c + b d)$

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Solution

Given: $(a + b) (c - d) + (a - b) (c + d) + 2 (a c + b d)$
Simplifying: $(a + b) (c - d) + (a - b) (c + d) + 2 (a c + b d)$
$= a (c - d) + b (c - d) + a (c + d) - b (c + d) + 2 (a c + b d)$
$= a c - a d + b c - b d + a c + a d - b c - b d + 2 a c + 2 b d$
$= a c + a c + 2 a c - a d + a d + b c - b c - b d - b d + 2 b d$
$= (a c + a c + 2 a c) + (- a d + a d) + (b c - b c) + (- b d - b d + 2 b d)$
$= 4 a c + 0 + 0 + 0$
$= 4 a c$
Therefore, $(a + b) (c - d) + (a - b) (c + d) + 2 (a c + b d)$ $= 4 a c$ .

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