Polynomial

We pick one term from each polynomial to try to "make" x^2.

$(-5x^3 \color{purple}- 5x^2 \color{blue}- 7x \color{green}+ 1\color{black})(\color{green}x^2 \color{blue}- 6x\color{purple} + 7\color{black})$

If we combine the colored terms with the same color together, we get terms in x^2. We add them up to get the total:

Term in x^2 = $\color{purple}(-5x^2)(+7)\color{black}+\color{blue}(-7x)(-6x)\color{black}+\color{green}(+1)(x^2)\color{black}=8x^2$

Therefore the coefficient of x^2 in the product is 8.