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Addition and Subtraction of Algebraic Expression

What must be ...

Question

What must be subtracted from $a^{2} b + 6 a^{2} b^{2} - 7 a b + 4$ to get $- 7 a^{2} b^{2} + 2 a^{3} b + 6 a b - 1$

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Solution

Let $x$ be the expression subtracted from $a^{2} b + 6 a^{2} b^{2} - 7 a b + 4$ to get $- 7 a^{2} b^{2} + 2 a^{3} b + 6 a b - 1$

$\Rightarrow a^{2} b + 6 a^{2} b^{2} - 7 a b + 4 - x = - 7 a^{2} b^{2} + 2 a^{3} b + 6 a b - 1$

$\Rightarrow - x = - 7 a^{2} b^{2} + 2 a^{3} b + 6 a b - 1 - (a^{2} b + 6 a^{2} b^{2} - 7 a b + 4)$

$\Rightarrow - x = - 7 a^{2} b^{2} + 2 a^{3} b + 6 a b - 1 - a^{2} b - 6 a^{2} b^{2} + 7 a b - 4$

$\Rightarrow - x = - 13 a^{2} b^{2} + 2 a^{3} b - a^{2} b + 13 a b - 5$

$∴ x = 13 a^{2} b^{2} - 2 a^{3} b + a^{2} b - 13 a b + 5$

$∴, 13 a^{2} b^{2} - 2 a^{3} b + a^{2} b - 13 a b + 5$ must be subtracted.

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