If $(x^{2} + 5 x + 7) (x^{2} - 5 x + 7)$ = $m^{2} - n^{2}$ , find the value of the sum of square of $m$ and $n$ .

x^{2} + 49 + 29 x^{2}

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x^{2} + 49 - 35 x

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x^{2} + 49 + 39 x^{2}

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x^{2} + 49 - 35 x

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Solution

The correct option is C $x^{2} + 49 + 39 x^{2}$
$(x^{2} + 5 x + 7) (x^{2} - 5 x + 7)$
= $(x^{2} + 7 + 5 x) (x^{2} + 7 - 5 x)$
= $[(x^{2} + 7) + (5 x)] [(x^{2} + 7) - (5 x)]$
= $(x^{2} + 7)^{2} - (5 x)^{2}$ ------(i)
But, according to the question, $(x^{2} + 5 x + 7) (x^{2} - 5 x + 7)$ = $m^{2} - n^{2}$ -----(ii)
From (i) and (ii), $m = x^{2} + 7$ and $n = 5 x$
Hence, $m^{2} + n^{2}$ = $(x^{2} + 7)^{2} + (5 x)^{2}$
= $x^{4} + 49 + 14 x^{2} + 25 x^{2}$
= $x^{4} + 49 + 39 x^{2}$

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