Which of the following experessions is equivalent to the expression given below ?
`(2x ^3 -x-1)-3(x^4+2x^3-2x^2-x-3)`

Options

A. `x^(14)-3`
B. `-3x^(14)`
C. `-3x^4+8x^3-6x^2-4x+8`
D. `-3x^4+4x^3-2x^2-2x-3`

Answer

To solve the expression \((2x^3 - x - 1) - 3(x^4 + 2x^3 - 2x^2 - x - 3)\), we will follow these steps:

### Step 1: Distribute the -3 across the terms in the parentheses.
We start with the expression:
\[ (2x^3 - x - 1) - 3(x^4 + 2x^3 - 2x^2 - x - 3) \]
Distributing -3 gives us:
\[ (2x^3 - x - 1) - (3x^4 + 6x^3 - 6x^2 + 3x + 9) \]

### Step 2: Rewrite the expression without parentheses.
Now we can rewrite the expression as:
\[ 2x^3 - x - 1 - 3x^4 - 6x^3 + 6x^2 - 3x - 9 \]

### Step 3: Combine like terms.
Now we will combine the like terms:
- For \(x^4\): \(-3x^4\)
- For \(x^3\): \(2x^3 - 6x^3 = -4x^3\)
- For \(x^2\): \(6x^2\)
- For \(x\): \(-x - 3x = -4x\)
- For the constant terms: \(-1 - 9 = -10\)

Putting it all together, we have:
\[ -3x^4 - 4x^3 + 6x^2 - 4x - 10 \]

### Step 4: Write the final expression.
Thus, the final expression is:
\[ -3x^4 - 4x^3 + 6x^2 + 2x + 8 \]

### Final Answer:
The expression equivalent to the given expression is:
\[ -3x^4 - 4x^3 + 6x^2 + 2x + 8 \]