If $x + y = 3$ and $x y = 6$ , what is the value of the expression $(x^{2} + y^{2})^{2} + 4 x^{2} y^{2} + 4 x y^{2} + 4 x^{2} y$

36

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45

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81

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72

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Solution

The correct option is C $81$
Given the expression
$(x^{2} + y^{2})^{2} + 4 x^{2} y^{2} + 4 y x^{2} + 4 x y^{2}$

On manipulating the terms of this expression we get,
$\Rightarrow (x^{2} + y^{2})^{2} + (2 x y)^{2} + (4 x y) (x^{2} + y^{2})$

$\Rightarrow (x^{2} + y^{2})^{2} + (2 x y)^{2} + 2 \times (2 x y) (x^{2} + y^{2})$

On comparing with the identity $(a + b)^{2} = a^{2} + b^{2} + 2 a b$ we get,

$\Rightarrow (x^{2} + y^{2})^{2} + (2 x y)^{2} + 2 \times (2 x y) (x^{2} + y^{2}) = [(x^{2} + y^{2}) + 2 x y]^{2}$

Again applying the identity $(a + b)^{2} = a^{2} + b^{2} + 2 a b$ we get,
$\Rightarrow [(x^{2} + y^{2}) + 2 x y]^{2} = [x^{2} + y^{2} + 2 x y]^{2} = [(x + y)^{2}]^{2} = (x + y)^{4}$

Now, putting value of $x + y = 3$ in the simplified equation we get,
$(x + y)^{4} = (3)^{4} = 81.$

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