Simplify
${(x - 2 y + 3)}^{2} + {(x + 2 y - 3)}^{2}$

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Solution

$(x - 2 y + 3)^{2} + (x + 2 y + 3)^{2}$
We know that,
$(a + b + c) = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a$
$∴ (x - 2 y + 3)^{2} + (x + 2 y + 3)^{2} =$
$x^{2} + 4 y^{2} + 9 - 4 x y - 12 y + 6 x + x^{2} + 4 y^{2} + 9 + 4 x y + 12 y + 6 x$
$= 2 (x^{2} + 4 y^{2} + 9 + 6 x)$
$= 2 x^{2} + 8 y^{2} + 18 + 12 x$

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(x - 2 y + 3)^{2} + (x + 2 y - 3)^{2}

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