What is the coefficient of $x^2$ when $-5x^3 - 5x^2 - 7x + 1$ is multiplied by $-x^2 - 6x + 1$ and the like terms are combined?
I want to find the coefficient of $x^2$ in the product of two polynomials, so I need to focus on the terms that produce $x^2$ when multiplied. To do that, I can use the distributive property and multiply each term in the first polynomial by each term in the second polynomial, and then look for the terms that have $x^2$ as a factor. So, multiplying the first term of the first polynomial by each term of the second polynomial, I get $(-5x^3)(-x^2) + (-5x^3)(-6x) + (-5x^3)(1)$. Simplifying, I get $5x^5 + 30x^4 - 5x^3$. None of these terms have $x^2$ as a factor, so I can ignore them for now. Moving on to the second term of the first polynomial, I get $(-5x^2)(-x^2) + (-5x^2)(-6x) + (-5x^2)(1)$. Simplifying, I get $5x^4 + 30x^3 - 5x^2$. Here, I see that the last term has $x^2$ as a factor, so I will keep track of it. The coefficient of $x^2$ in this term is $-5$. Moving on to the third term of the first polynomial, I get $(-7x)(-x^2) + (-7x)(-6x) + (-7x)(1)$. Simplifying, I get $7x^3 + 42x^2 - 7x$. Here, I see that the second term has $x^2$ as a factor, so I will keep track of it as well. The coefficient of $x^2$ in this term is $42$. Finally, multiplying the last term of the first polynomial by each term of the second polynomial, I get $(1)(-x^2) + (1)(-6x) + (1)(1)$. Simplifying, I get $-x^2 - 6x + 1$. Here, I see that the first term has $x^2$ as a factor, so I will keep track of it too. The coefficient of $x^2$ in this term is $-1$. Now, I have found all the terms that have $x^2$ as a factor in the product of the two polynomials. To find the coefficient of $x^2$ in the product, I need to add the coefficients of $x^2$ in these terms. That is, the coefficient of $x^2$ in the product is $-5 + 42 - 1 = 36$. 