Certainly! Let's carefully break down and analyze the given mathematical expression step-by-step:
The expression we need to consider is:
[tex]\[
-m n\left(m^2-2 n\right)^2
\][/tex]
### Step-by-Step Breakdown
1. Identifying the Components:
- The expression consists of three primary components:
- A negative sign (`-`).
- The product of `m` and `n`.
- The square of the term [tex]\((m^2 - 2n)\)[/tex].
2. Understanding [tex]\((m^2 - 2n)^2\)[/tex]:
- The term inside the parentheses is [tex]\(m^2 - 2n\)[/tex].
- Squaring this term gives us [tex]\((m^2 - 2n)^2\)[/tex].
- Note that [tex]\((m^2 - 2n)^2\)[/tex] means [tex]\((m^2 - 2n)\)[/tex] multiplied by itself.
3. Constructing the Expression:
- The next step is to handle the multiplication outside the squared term.
- We have the negative sign [tex]\( - \)[/tex] and the product [tex]\(m n\)[/tex].
- Combine these to form [tex]\(-m n\)[/tex].
4. Combining All Parts:
- Multiply the term [tex]\(-m n\)[/tex] by the squared term:
- The complete expression is [tex]\(-m n \left(m^2 - 2n\right)^2\)[/tex].
Thus, the simplified form of the expression remains:
[tex]\[
-m n\left(m^2-2 n\right)^2
\][/tex]
This is already in its simplified form and does not require further simplification. As demonstrated in the steps above, every component of the expression has been precisely identified, and combining them results in the given expression.
