To add the polynomial expressions \(3x^4 - 2 - 2x^3 + 4x\) and \(2x^3 - 6x - 4\), we first need to combine like terms.
Let's rewrite the expressions clearly:
Expression #1:
\[3x^4 - 2 - 2x^3 + 4x\]
Expression #2:
\[2x^3 - 6x - 4\]
Now we can combine them:
-
Combine the \(x^4\) terms:
- From Expression #1, we have \(3x^4\).
- (There's no \(x^4\) term in Expression #2, so it remains \(3x^4\).)
-
Combine the \(x^3\) terms:
- From Expression #1, we have \(-2x^3\).
- From Expression #2, we have \(+2x^3\).
- So, \(-2x^3 + 2x^3 = 0\).
-
Combine the \(x\) terms:
- From Expression #1, we have \(+4x\).
- From Expression #2, we have \(-6x\).
- So, \(4x - 6x = -2x\).
-
Combine the constant terms:
- From Expression #1, we have \(-2\).
- From Expression #2, we have \(-4\).
- So, \(-2 - 4 = -6\).
Now we can write the combined polynomial:
\[ 3x^4 + 0 - 2x - 6 \text{ simplifies to } 3x^4 - 2x - 6. \]
So the final result is:
\[ \boxed{A) , 3x^4 - 2x - 6} \]