−𝟕𝟏𝒂𝟑𝒃 − 𝟖𝟐𝒂𝟒𝒃
𝟐 + 𝟓𝟎𝒂𝟑𝒃 + 𝟖𝟒𝒂
𝟒𝒃
𝟐 + 𝟒𝟓𝒂
𝟑�

It looks like there might be a formatting issue in the expression you provided. It seems like some terms might have been split or misplaced. However, I'll try to interpret the expression and simplify it.

The expression appears to be:

\[
-71a^3b - \frac{82a^4b}{2} + 50a^3b + \frac{84a^4b}{2} + 45a^3
\]

Let's simplify it step by step:

### Step 1: Simplify the fractions
Simplify the fractional terms first:

\[
\frac{82a^4b}{2} = 41a^4b
\]
\[
\frac{84a^4b}{2} = 42a^4b
\]

Substitute these back into the expression:

\[
-71a^3b - 41a^4b + 50a^3b + 42a^4b + 45a^3
\]

### Step 2: Combine like terms
Group and combine like terms:

1. For the \(a^4b\) terms:
   \[
   -41a^4b + 42a^4b = 1a^4b = a^4b
   \]

2. For the \(a^3b\) terms:
   \[
   -71a^3b + 50a^3b = -21a^3b
   \]

3. The \(a^3\) term:
   \[
   +45a^3
   \]

### Final Expression

So, the simplified expression is:

\[
\boxed{a^4b - 21a^3b + 45a^3}
\]

Would you like more details or have any questions?

Here are 8 related questions you might consider:
1. How would the expression change if the coefficients were negative fractions?
2. What if the \(a^3\) term also had a \(b\) term attached to it?
3. How would you factor out the greatest common factor from the simplified expression?
4. What happens if one of the terms includes a different variable, like \(c\)?
5. How would the expression simplify if \(a = 1\) and \(b = 1\)?
6. What if the expression included higher powers, such as \(a^5\)?
7. How does the simplification process differ when working with polynomials with more variables?
8. How can you verify the correctness of the simplified expression?

**Tip:** Always combine like terms to simplify polynomial expressions. This makes it easier to understand and work with the expression.

Learn how to simplify the polynomial expression -71a^3b - 82a^4b/2 + 50a^3b + 84a^4b/2 + 45a^3 step by step. This problem involves combining like terms and simplifying fractions in polynomials. Suitable for high school students studying algebra.

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