What should be subtracted from $x^{2} - x y + y^{2} - x + y + 3$ to obtain $- x^{2} + 3 y^{2} - 4 x y + 1 ?$

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Solution

Let ‘E’ be the required expression. Then, we have

$x^{2} - x y + y^{2} - x + y + 3 - E = - x^{2} + 3 y^{2} - 4 x y + 1$

Therefore, $E = (x^{2} - x y + y^{2} - x + y + 3) - (- x^{2} + 3 y^{2} - 4 x y + 1)$

$= x^{2} - x y + y^{2} - x + y + 3 + x^{2} - 3 y^{2} + 4 x y - 1$

Collecting positive and negative like terms together, we get

$x^{2} + x^{2} - x y + 4 x y + y^{2} - 3 y^{2} - x + y + 3 - 1$

$= 2 x^{2} + 3 x y - 2 y^{2} - x + y + 2$

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