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?
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{align*}
(-5x^2)(1)+(-7x)(-6x)+(1)(-x^2)&=-5x^2+42x^2-x^2\\
&=\boxed{36}x^2
\end{align*}Consequently, the desired coefficient is $\boxed{36}$.
36
Level 4
Algebra
Problem: 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?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the coefficient of \(x^2\) in the product of the two polynomials, we need to consider how the terms from each polynomial can combine to produce \(x^2\). 

The first polynomial is \(-5x^3 - 5x^2 - 7x + 1\) and the second polynomial is \(-x^2 - 6x + 1\). We will identify the pairs of terms from each polynomial that, when multiplied, yield a term of degree 2.

1. The term \(-5x^2\) from the first polynomial can be multiplied by the constant term \(1\) from the second polynomial:
   \[
   -5x^2 \cdot 1 = -5x^2
   \]

2. The term \(-7x\) from the first polynomial can be multiplied by the linear term \(-6x\) from the second polynomial:
   \[
   -7x \cdot -6x = 42x^2
   \]

3. The constant term \(1\) from the first polynomial can be multiplied by the quadratic term \(-x^2\) from the second polynomial:
   \[
   1 \cdot -x^2 = -x^2
   \]

Now, we combine all the contributions to the \(x^2\) term:
\[
-5x^2 + 42x^2 - x^2
\]

Calculating this step-by-step:
- Start with \(-5x^2\).
- Add \(42x^2\):
  \[
  -5x^2 + 42x^2 = 37x^2
  \]
- Then subtract \(x^2\):
  \[
  37x^2 - x^2 = 36x^2
  \]

Thus, the coefficient of \(x^2\) in the product of the two polynomials is \(36\). Therefore, the final answer is \(36\).<|END_OF_LATENT|> The final answer is $36$.