To simplify the expression [tex]\(\frac{-14 m^2 + 4 m n - m^3 n^2}{2 m n}\)[/tex], follow these steps:
1. Write down the original expression:
[tex]\[
\frac{-14 m^2 + 4 m n - m^3 n^2}{2 m n}
\][/tex]
2. Break down the numerator and distribute the denominator:
Every term in the numerator can be divided individually by the denominator [tex]\(2 m n\)[/tex].
[tex]\[
\frac{-14 m^2}{2 m n} + \frac{4 m n}{2 m n} - \frac{m^3 n^2}{2 m n}
\][/tex]
3. Simplify each term individually:
- For the first term:
[tex]\[
\frac{-14 m^2}{2 m n} = \frac{-14 m^2}{2 m n} = -7 \frac{m^2}{m n} = -7 \frac{m}{n}
\][/tex]
- For the second term:
[tex]\[
\frac{4 m n}{2 m n} = \frac{4 m n}{2 m n} = \frac{4}{2} = 2
\][/tex]
- For the third term:
[tex]\[
\frac{m^3 n^2}{2 m n} = \frac{m^3 n^2}{2 m n} = \frac{m^{3-1} n^{2-1}}{2} = \frac{m^2 n}{2}
\][/tex]
4. Rewrite the simplified terms together:
- From the first term, we have [tex]\(-7 \frac{m}{n}\)[/tex]
- From the second term, we have [tex]\(2\)[/tex]
- From the third term, we have [tex]\(\frac{m^2 n}{2}\)[/tex]
[tex]\[
\frac{-14 m^2 + 4 m n - m^3 n^2}{2 m n} = -7 \frac{m}{n} + 2 - \frac{m^2 n}{2}
\][/tex]
5. Combine the simplified terms into one expression:
[tex]\[
-\frac{m^2 n}{2} - 7 \frac{m}{n} + 2
\][/tex]
So, the final simplified expression is:
[tex]\[
-\frac{m^2 n}{2} - \frac{7 m}{n} + 2
\][/tex]
