jmc

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: $$\begin{array}{rl}& (6x^3)(4)+(4x^2)(3x)+(-7x)(3x^2)+(-5)(6x^3)\\ & =24x^3+12x^3-21x^3-30x^3\\ & =\boxed{-15}{x}^3\end{array}$$