The values of $k$ for which $x^{2} + 5 k x + k^{2} + 5$ is exactly divisible by $x + 2$ but not by $x + 3$ is (are)

1

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5

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1, 9

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9

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Solution

The correct option is D $9$
$P (x) = x^{2} + 5 k x + k^{2} + 5$
$P (- 2) = 0$
$4 - 10 k + k^{2} + 5 = 0$
$k = 9, 1$ .....(1)
But $P (- 3) = 0$
$9 - 15 k + k^{2} + 5 \neq 0$
$(k - 14) (k - 1) \neq 0$
$k \neq 14, 1$ ....(2)
from (1) and (2) $k = 9$

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