$p^{3} q^{3} - p q - p^{2} q^{2} + p^{4} q^{4}$ can be regrouped and simplified as _______.

(p q) (1 + p q) (1 - p q)^{2}

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(- p q) (1 - p q) (1 + p q)^{2}

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(p q) (1 + p q) (1 + p q)^{2}

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(- p q) (1 - p q) (1 + p q)^{2}

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Solution

The correct option is D $(- p q) (1 - p q) (1 + p q)^{2}$
Given: $p^{3} q^{3} - p q - p^{2} q^{2} + p^{4} q^{4}$

$p^{3} q^{3} - p q - p^{2} q^{2} + p^{4} q^{4} \Rightarrow p q (p^{2} q^{2} - 1) - p^{2} q^{2} (1 - p^{2} q^{2}) \Rightarrow (- p q) (1 - p^{2} q^{2}) - p^{2} q^{2} (1 - p^{2} q^{2}) \Rightarrow (1 - p^{2} q^{2}) (- 1) (p q + p^{2} q^{2}) \Rightarrow (1 - p^{2} q^{2}) (- p q) (1 + p q) \Rightarrow (1 + p q) (1 - p q) (- p q) (1 + p q) \Rightarrow (- p q) (1 - p q) (1 + p q)^{2}$

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