3*r*−1⋅[2(*r*−1+2)−*r*−1(2−*r*−1)−4]

Let's simplify the expression:

\[
3r - 1 \cdot \left[ 2(r - 1 + 2) - r - 1 \cdot (2 - r - 1) - 4 \right]
\]

### Step 1: Simplify the terms inside the brackets.

First, simplify \( r - 1 + 2 \):
\[
r - 1 + 2 = r + 1
\]
So the expression becomes:
\[
3r - 1 \cdot \left[ 2(r + 1) - (r - 1)(2 - r - 1) - 4 \right]
\]

### Step 2: Expand \( 2(r + 1) \):
\[
2(r + 1) = 2r + 2
\]
Now the expression is:
\[
3r - 1 \cdot \left[ 2r + 2 - (r - 1)(2 - r - 1) - 4 \right]
\]

### Step 3: Simplify \( (r - 1)(2 - r - 1) \).
First, simplify \( 2 - r - 1 \):
\[
2 - r - 1 = 1 - r
\]
Now expand \( (r - 1)(1 - r) \):
\[
(r - 1)(1 - r) = -r + r^2 - 1
\]
So the expression becomes:
\[
3r - 1 \cdot \left[ 2r + 2 - (-r + r^2 - 1) - 4 \right]
\]

### Step 4: Simplify inside the brackets:
\[
2r + 2 - (-r + r^2 - 1) = 2r + 2 + r - r^2 + 1 = 3r - r^2 + 3
\]
Now the expression is:
\[
3r - 1 \cdot \left[ 3r - r^2 + 3 - 4 \right]
\]

Simplify further:
\[
3r - r^2 - 1
\]
So the expression becomes:
\[
3r - 1 \cdot (3r - r^2 - 1)
\]

### Step 5: Multiply by \( -1 \):
\[
3r - (3r - r^2 - 1) = 3r - 3r + r^2 + 1
\]

### Step 6: Simplify:
\[
r^2 + 1
\]

The simplified expression is:
\[
r^2 + 1
\]

Would you like more details or have any questions?

### Related Questions:
1. How would this expression change if a different constant was used in place of 3r?
2. What happens if \( r = 0 \) or \( r = 1 \)?
3. How can this expression be represented graphically?
4. What are the implications of simplifying expressions involving multiple variables?
5. How can the properties of exponents simplify terms in similar expressions?

**Tip:** When simplifying expressions, always work through parentheses and brackets first, following the order of operations.

Learn how to simplify the algebraic expression 3r − 1 ⋅ [2(r − 1 + 2) − r − 1(2 − r − 1) − 4] step-by-step. This solution covers key concepts in algebra, such as polynomial expansion and simplification. Suitable for students in Grades 9-12.

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