Sure, let's solve the problem step by step using the expressions provided.
### Step 1: Identifying Expressions
The given expressions are:
1. [tex]\(m - 7mn\)[/tex]
2. [tex]\(-3mn^2\)[/tex]
3. [tex]\(m + 3(m - 11)^2\)[/tex]
4. [tex]\(m(-8m)\)[/tex]
5. [tex]\(m(2mn) - 6m\)[/tex]
6. [tex]\(m + 3n^2\)[/tex]
### Step 2: Find [tex]\( \text{ the product of } m \text{ and a multiple of } m\)[/tex]
A product that involves both [tex]\(m\)[/tex] and [tex]\(m\)[/tex] (a multiple of [tex]\(m\)[/tex]) should have the form [tex]\(m \times (\text{something with } m)\)[/tex].
From the given expressions, [tex]\( m(2mn) - 6m \)[/tex] is such a product. It involves [tex]\(m\)[/tex] multiplied by [tex]\(2mn\)[/tex], which is a term that depends on [tex]\(m\)[/tex]:
[tex]\[ m(2mn) - 6m \][/tex]
Thus, the product of [tex]\(m\)[/tex] and a multiple of [tex]\(m\)[/tex] is:
[tex]\[ m(2mn) - 6m \][/tex]
### Step 3: Find [tex]\( \text{ the product of } m \text{ and a factor not depending on } m\)[/tex]
A product of [tex]\(m\)[/tex] and something that doesn't include [tex]\(m\)[/tex] should have the form [tex]\(m \times (\text{something without } m)\)[/tex].
Considering the simplest form among the expressions, [tex]\( -3mn^2 \)[/tex] has [tex]\( -3n^2 \)[/tex] which does not depend on [tex]\(m\)[/tex]:
[tex]\[ -3n^2 \][/tex]
Hence, the product of [tex]\(m\)[/tex] and a factor not depending on [tex]\(m\)[/tex] is:
[tex]\[ -3n^2 \][/tex]
### Step 4: Find [tex]\( \text{ the sum of } m \text{ and a term depending on } m \)[/tex]
A sum that includes [tex]\(m\)[/tex] plus a term that also depends on [tex]\(m\)[/tex] (or any other variable) should fit the form [tex]\(m + (\text{other term})\)[/tex].
From the given expressions, [tex]\( m + 3n^2 \)[/tex] fits this description as it is [tex]\(m\)[/tex] plus another term, [tex]\(3n^2\)[/tex]:
[tex]\[ m + 3n^2 \][/tex]
So, the sum of [tex]\(m\)[/tex] and a term depending on [tex]\(m\)[/tex] is:
[tex]\[ m + 3n^2 \][/tex]
### Final Result:
- The product of [tex]\(m\)[/tex] and a multiple of [tex]\(m\)[/tex]: [tex]\( m(2mn) - 6m \)[/tex]
- The product of [tex]\(m\)[/tex] and a factor not depending on [tex]\(m\)[/tex]: [tex]\( -3n^2 \)[/tex]
- The sum of [tex]\(m\)[/tex] and a term depending on [tex]\(m\)[/tex]: [tex]\( m + 3n^2 \)[/tex]
Thus, the solutions are:
[tex]\[ \boxed{m(2mn) - 6m} \][/tex]
[tex]\[ \boxed{-3n^2} \][/tex]
[tex]\[ \boxed{m + 3n^2} \][/tex]
