Which of the following is equivalent to `(2b^(3)c^(2)+b^(2)c)-(b^(3)c^(2)-b^(2)c-4bc)`?

Options

A. `0`
B. `b^(3)c^(2)`
C. `b^(3)c^(2)+2b^(2)c`
D. `b^(3)c^(2)+2b^(2)c-8bc`

Explanation

To solve the expression \( (2b^{3}c^{2} + b^{2}c) - (b^{3}c^{2} - b^{2}c - 4bc) \), we will follow these steps:

### Step 1: Write down the expression
We start with the expression:
\[ (2b^{3}c^{2} + b^{2}c) - (b^{3}c^{2} - b^{2}c - 4bc) \]

### Step 2: Distribute the negative sign
Next, we distribute the negative sign across the second parenthesis:
\[ 2b^{3}c^{2} + b^{2}c - b^{3}c^{2} + b^{2}c + 4bc \]

### Step 3: Combine like terms
Now, we will combine the like terms:
- For \( b^{3}c^{2} \): \( 2b^{3}c^{2} - b^{3}c^{2} = (2 - 1)b^{3}c^{2} = 1b^{3}c^{2} \)
- For \( b^{2}c \): \( b^{2}c + b^{2}c = 2b^{2}c \)
- For \( bc \): \( -4bc + 4bc = 0 \)

Putting it all together, we have:
\[ b^{3}c^{2} + 2b^{2}c + 0 \]

### Step 4: Write the final answer
Thus, the final simplified expression is:
\[ b^{3}c^{2} + 2b^{2}c \]