Certainly! Let's go through the given expression step by step:
The original expression to simplify is:
[tex]\[
\left(4 m^2 + 14 m^3 n\right) - \left(-m n^3 - 4 m^3 n - 11 m^2\right) - \left(m n^3 - 13 m^2\right)
\][/tex]
### Step 1: Distribute the negative signs
First, let's distribute the negative signs across the terms inside the parentheses.
[tex]\[
\left(4 m^2 + 14 m^3 n\right) - \left(-m n^3 - 4 m^3 n - 11 m^2\right) - \left(m n^3 - 13 m^2\right)
\][/tex]
Becomes:
[tex]\[
4 m^2 + 14 m^3 n + m n^3 + 4 m^3 n + 11 m^2 - m n^3 + 13 m^2
\][/tex]
### Step 2: Combine like terms
Now, we combine all like terms. The like terms include [tex]\(m^2\)[/tex], [tex]\(m^3n\)[/tex], and [tex]\(mn^3\)[/tex]:
- [tex]\(m^2\)[/tex] terms: [tex]\(4 m^2 + 11 m^2 + 13 m^2\)[/tex]
- [tex]\(m^3 n\)[/tex] terms: [tex]\(14 m^3 n + 4 m^3 n\)[/tex]
- [tex]\(m n^3\)[/tex] terms: [tex]\(m n^3 - m n^3\)[/tex]
Combining these, we get:
[tex]\[
(4 + 11 + 13)m^2 + (14 + 4) m^3 n + (1 - 1) m n^3
\][/tex]
Simplifying each group of terms, we get:
[tex]\[
28 m^2 + 18 m^3 n + 0 m n^3
\][/tex]
### Step 3: Simplify the expression
Since [tex]\(0 m n^3\)[/tex] is zero, we can remove that term, leaving us with:
[tex]\[
28 m^2 + 18 m^3 n
\][/tex]
### Step 4: Factor out the greatest common factor
Lastly, we factor out the greatest common factor from [tex]\(28 m^2 + 18 m^3 n\)[/tex]. The greatest common factor is [tex]\(2 m^2\)[/tex]:
[tex]\[
28 m^2 + 18 m^3 n = 2 m^2 (14) + 2 m^2 (9 m n)
\][/tex]
Thus, the simplified form is:
[tex]\[
10 m^2 (m n - 2)
\][/tex]
### Conclusion
The simplified expression is:
[tex]\[
\boxed{10 m^2 (m n - 2)}
\][/tex]
