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: $$(6x^3)(4)+(4x^2)(3x)+(-7x)(3x^2)+(-5)(6x^3).$$We simplify to find: \begin{align*} &(6x^3)(4)+(4x^2)(3x)+(-7x)(3x^2)+(-5)(6x^3)\\ &\qquad=24x^3+12x^3-21x^3-30x^3\\ &\qquad=\boxed{-15}x^3 \end{align*}