Let's simplify the expression \( -2x^3 + 2x^2 - 3x + x^2 + 4 - x - 3x^2 \) step by step by combining like terms.
### Step 1: Group and Identify Like Terms
Group the terms involving the same powers of \(x\):
1. Terms involving \(x^3\): \( -2x^3 \)
2. Terms involving \(x^2\): \( 2x^2 + x^2 - 3x^2 \)
3. Terms involving \(x\): \( -3x - x \)
4. Constant terms: \( 4 \)
### Step 2: Combine Like Terms
1. For \(x^3\):
[tex]\[
-2x^3
\][/tex]
2. For \(x^2\):
[tex]\[
2x^2 + x^2 - 3x^2 = 2x^2 + 1x^2 - 3x^2 = 3x^2 - 3x^2 = 0
\][/tex]
3. For \(x\):
[tex]\[
-3x - x = -4x
\][/tex]
4. Constant term remains the same:
[tex]\[
4
\][/tex]
### Step 3: Write the Simplified Expression
Combine all the simplified terms:
[tex]\[
-2x^3 + 0x^2 - 4x + 4
\][/tex]
Since \(0x^2\) does not contribute anything, we can omit it:
[tex]\[
-2x^3 - 4x + 4
\][/tex]
So, the simplified expression is \(-2x^3 - 4x + 4\).
### Step 4: Match with the Given Options
Now, let's match the simplified expression with the given options:
1. \( -2x^2 + x + 4 \)
2. \( -2x^3 + x^2 - 4x + 4 \)
3. \( -2x^3 - 4x + 4 \)
4. \( -2x^2 - 4x + 4 \)
The simplified expression \(-2x^3 - 4x + 4\) matches option 3.
### Conclusion
The correct option is:
[tex]\[
\boxed{-2x^3 - 4x + 4}
\][/tex]
