Let P ( x ) be a real polynomial of degree 3 which vanishes at x =...

Let ${P}^{'} ( x ) = a ( x - 1 ) ( x + 1 ) = a\left({x}^{2} - 1\right)$

$P\left(x\right) = a ∫ \left({x}^{2} - 1\right)dx + c$

$= a\left(\dfrac{{x}^{3}}{3} - x\right) + c$

Now $P ( - 3 ) = 0$

$⇒ a\left(- \dfrac{27}{3} + 3\right) + c = 0$

$⇒ - 6a + c = 0 \ldots ( 1 )$

Now ${∫}_{- 1}^{1}\left(a\left(\dfrac{{x}^{3}}{3} - x\right) + c\right)dx = 18$

$= 2c = 18 ⇒ c = 9 . . . \left(2\right)$

From $( 1 ) \text{ and } ( 2 )$ $⇒ - 6a + 9 = 0 ⇒ a = \dfrac{3}{2}$

$⇒ P\left(x\right) = \dfrac{3}{2}\left(\dfrac{{x}^{3}}{3} - x\right) + 9$

Then, the Sum of coefficient$= \dfrac{1}{2} - \dfrac{3}{2} + 9$

$= 8$