To multiply terms with exponents, the base of both exponents must be the same. In this case they are so multiply the coefficients and add the exponents:

-4z⁴ x 3z⁴
(-4 x 3)z^{(4 + 4)}
-12z⁸

The ratio of football cards to baseball cards is 3:2 and the ratio of baseball cards to basketball cards is 3:1. To solve this problem, we need the baseball card side of each ratio to be equal so we need to rewrite the ratios in terms of a common number of baseball cards. (Think of this like finding the common denominator when adding fractions.) The ratio of football to baseball cards can also be written as 9:6 and the ratio of baseball cards to basketball cards as 6:2. So, the ratio of football cards to basketball cards is football:baseball, baseball:basketball or 9:6, 6:2 which reduces to 9:2.

The factors of 40 are [1, 2, 4, 5, 8, 10, 20, 40] and the factors of 40 are [1, 2, 4, 5, 8, 10, 20, 40]. They share 8 factors [1, 2, 4, 5, 8, 10, 20, 40] making 40 the greatest factor 40 and 40 have in common.

First, solve for \( \left|c + 8\right| \):

\( \left|c + 8\right| \) - 1 = 5
\( \left|c + 8\right| \) = 5 + 1
\( \left|c + 8\right| \) = 6

The value inside the absolute value brackets can be either positive or negative so (c + 8) must equal + 6 or -6 for \( \left|c + 8\right| \) to equal 6:

So, c = -14 or c = -2.

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 5 are [5, 10, 15, 20, 25, 30, 35, 40, 45, 50] and the first few multiples of 7 are [7, 14, 21, 28, 35, 42, 49, 56, 63, 70]. The first few multiples they share are [35, 70] making 35 the smallest multiple 5 and 7 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{6 x 7}{5 x 7} \) + \( \frac{6 x 5}{7 x 5} \)

\( \frac{42}{35} \) + \( \frac{30}{35} \)

Now, because the fractions share a common denominator, you can add them:

\( \frac{42 + 30}{35} \) = \( \frac{72}{35} \) = 2\(\frac{2}{35}\)