Sure! Let's go step-by-step to simplify the given expression:
[tex]\(7mn(2m^2 - 4n^2 + 2) + mn(m^2 - n^2)\)[/tex]
### Step 1: Distribute [tex]\(mn\)[/tex] inside the parentheses
First, distribute [tex]\(7mn\)[/tex] inside the first parentheses:
[tex]\[7mn(2m^2) - 7mn(4n^2) + 7mn(2)\][/tex]
[tex]\[= 14m^3n - 28mn^3 + 14mn\][/tex]
Now distribute [tex]\(mn\)[/tex] inside the second parentheses:
[tex]\[mn(m^2) - mn(n^2)\][/tex]
[tex]\[= m^3n - mn^3\][/tex]
### Step 2: Combine like terms
Now we have:
[tex]\[14m^3n - 28mn^3 + 14mn + m^3n - mn^3\][/tex]
Combine the terms:
1. [tex]\(m^3n\)[/tex] terms:
[tex]\[14m^3n + m^3n = 15m^3n\][/tex]
2. [tex]\(mn^3\)[/tex] terms:
[tex]\[-28mn^3 - mn^3 = -29mn^3\][/tex]
3. [tex]\(mn\)[/tex] terms:
[tex]\[14mn\][/tex]
### Step 3: Write the final simplified expression:
[tex]\[
15m^3n - 29mn^3 + 14mn
\][/tex]
So, the simplified form of the given expression is:
[tex]\[
15m^3n - 29mn^3 + 14mn
\][/tex]
