jmc

Instead of expanding the entire product, we can look only at terms that will multiply to give $x^2$. We know that: $$x^2=x^2\cdot 1=x\cdot x=1\cdot x^2$$Knowing this, the $x^2$ term in the expansion will be the sum of these three terms: $$(-5x^2)(1)+(-7x)(-6x)+(1)(-x^2)$$Simplifying gives: $$\begin{array}{rl}(-5x^2)(1)+(-7x)(-6x)+(1)(-x^2)& =-5x^2+42x^2-x^2\\ & =\boxed{36}{x}^2\end{array}$$Consequently, the desired coefficient is $\boxed{36}$.