A student simplifies $$ (6 b+4 r)-(2 b+r) $$ and says that the result is $$ 4 b+5 r $$. Explain the error and describe the correct steps to simplify the expression.

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The student made an error in distributing the subtraction across the terms inside the second set of parentheses. Specifically, they incorrectly **added** the $$ r $$ term instead of **subtracting** it. Here's a step-by-step explanation of the correct simplification process:

### Given Expression:
$$
(6b + 4r) - (2b + r)
$$

### Step 1: Distribute the Subtraction
When you subtract a parenthetical expression, you must distribute the negative sign to each term inside the parentheses:

$$
6b + 4r - 2b - r
$$

**Error Explanation:**  
The student subtracted $$ 2b $$ correctly but mistakenly **added** $$ r $$ instead of subtracting it, leading to $$ 4b + 5r $$.

### Step 2: Combine Like Terms
Now, combine the coefficients of like terms ($$ b $$ terms and $$ r $$ terms):

$$
(6b - 2b) + (4r - r) = 4b + 3r
$$

### Correct Simplified Expression:
$$
4b + 3r
$$

### Summary of the Error:
- **Incorrect Step by Student:**  
  $$
  (6b + 4r) - (2b + r) = 6b - 2b + 4r + r = 4b + 5r
  $$
  Here, the student incorrectly **added** $$ r $$ instead of subtracting it.

- **Correct Step:**  
  $$
  (6b + 4r) - (2b + r) = 6b - 2b + 4r - r = 4b + 3r
  $$

By properly distributing the negative sign to each term inside the parentheses, the correct simplified expression is $$ 4b + 3r $$.
The student incorrectly added the $$ r $$ term instead of subtracting it, resulting in $$ 4b + 5r $$. The correct simplification is $$ 4b + 3r $$.

A student simplifies \( (6 b+4 r)-(2 b+r) \) and says that the result is \( 4 b+5 r \). Explain the error and describe the correct steps to simplify the expression.

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