Simplify: `2a - [3b - {a - (2c - 3b) + 4c - 3 (a - b - 2c)}]`

Options

A. `3b + 8c `
B. `2b + 8c `
C. `3b + 15c `
D. `3b + 7c `

Explanation

To simplify the expression \( 2a - [3b - \{a - (2c - 3b) + 4c - 3(a - b - 2c)\}] \), we will follow these steps:

### Step 1: Start with the original expression
We have:
\[ 2a - [3b - \{a - (2c - 3b) + 4c - 3(a - b - 2c)\}] \]

### Step 2: Simplify the innermost brackets
First, we simplify \( a - (2c - 3b) \):
\[ a - (2c - 3b) = a - 2c + 3b \]
Now, substitute this back into the expression:
\[ 2a - [3b - \{(a - 2c + 3b) + 4c - 3(a - b - 2c)\}] \]

### Step 3: Expand the expression inside the outer brackets
Next, we simplify \( 3(a - b - 2c) \):
\[ 3(a - b - 2c) = 3a - 3b - 6c \]
Now substitute this back into the expression:
\[ 2a - [3b - \{(a - 2c + 3b) + 4c - (3a - 3b - 6c)\}] \]

### Step 4: Combine the terms inside the brackets
Now we simplify:
\[ (a - 2c + 3b) + 4c - (3a - 3b - 6c) = a - 2c + 3b + 4c - 3a + 3b + 6c \]
Combine like terms:
\[ = (a - 3a) + (3b + 3b) + (-2c + 4c + 6c) = -2a + 6b + 8c \]
Now substitute this back into the expression:
\[ 2a - [3b - (-2a + 6b + 8c)] \]

### Step 5: Distribute the negative sign
Distributing the negative sign gives:
\[ 2a - [3b + 2a - 6b - 8c] \]
This simplifies to:
\[ 2a - 3b - 2a + 6b + 8c \]

### Step 6: Combine like terms
Now combine the terms:
\[ (2a - 2a) + (-3b + 6b) + 8c = 0 + 3b + 8c \]
Thus, we have:
\[ 3b + 8c \]

### Final Answer
The simplified expression is:
\[ \boxed{3b + 8c} \]