Instead of expanding the entire product, we can look only at terms that will multiply to give $ x^3 $. We know that: $$x^3=x^3\cdot 1=x^2\cdot x=x\cdot x^2=1\cdot x^3$$Knowing this, the $ x^3 $ term in the expansion will be the sum of these four terms: $$(-3x^3)(4)+(-3x^2)(3x)+(-8x)(-7x^2)+(1)(2x^3)$$We simplify to find: \begin{align*} &(-3x^3)(4)+(-3x^2)(3x)+(-8x)(-7x^2)+(1)(2x^3)\\ &\qquad=-12x^3-9x^3+56x^3+2x^3\\ &\qquad=\boxed{37}x^3 \end{align*}