Let P ( x ) be a real polynomial of degree 3 which vanishes at x = - 3 . Let P ( x ) have local minima at x…

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

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

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

Now $P(-3)=0$

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

$\Rightarrow -6a+c=0 \dots \left(1\right)$

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

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

 From $\left(1\right) \& \left(2\right)$ $\Rightarrow -6a+9=0\Rightarrow a=\frac{3}{2}$

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

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

$=8$