Question

1. factor $-60a^{3}bc^{2} + 70a^{2}c - 70a^{2} - 20a$.

a. $10a(-6a^{2}bc^{2} + 7ac - 7a - 2)$
b. $a(60a^{2}bc^{2} - 70ac + 70 - 20)$
c. $10a(-6abc^{2} + 7ac - 7a - 2)$
d. $-60abc^{2} + 70ac - 70a - 20a$

Explanation:

Step1: Identify greatest common factor

The terms are $-60a^3bc^2$, $70a^2c$, $-70a^2$, $-20a$. The GCF of coefficients $60,70,70,20$ is $10$, and the GCF of variable parts is $a$. So overall GCF is $10a$.

Step2: Factor out the GCF

Divide each term by $10a$:
$\frac{-60a^3bc^2}{10a} = -6a^2bc^2$, $\frac{70a^2c}{10a}=7ac$, $\frac{-70a^2}{10a}=-7a$, $\frac{-20a}{10a}=-2$
Combine to get: $10a(-6a^2bc^2 + 7ac - 7a - 2)$

Step3: Verify other options

Option b has incorrect sign changes; option c incorrectly reduces $a^3$ to $a$; option d is the original expression, not factored.

Answer:

a. $10a(-6a^{2}bc^{2} + 7ac - 7a - 2)$