$f (x) = 3 x^{4} + 17 x^{3} + 9 x^{2} - 7 x - 10; g (x) = x + 5$

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Solution

According to the problem:-
Given that:
$f (x) = 3 x^{4} + 17 x^{3} + 9 x^{2} - 7 x - 10$
and
$g (x) = x + 5$

Using the factor theorem, $g (x)$ is a factor of polynomial $f (x)$

Therefore,
$x + 5 = 0$
$x = - 5$

Hence,
$f (- 5) = 3 (- 5)^{4} + 17 (- 5)^{3} + 9 (- 5)^{2} - 7 (5) - 10$

Implies that
$= 3 \times 625 + 17 \times (- 125) + 225 + 35 - 10$

Implies that
$= 1875 - 2125 + 250$
$= 0$

Hence the $g (x)$ is the factor of polynomial $f (x)$

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