$W\left(x\right)=-x^4-2x^3+ax+3$  

$W\left(-2\right)=-1$  

$W\left(-2\right)=-16-2\cdot \left(-8\right)-2a+3=-16+16-2a+3=-1$  

$-2a=-4$  

$\underline{a=2}$