To simplify the given expression, \(\left(5 x^2 - 3 x^3 + 4 x^4 + 7 x + 5\right) - \left(7 x + 5 - 3 x^3 + 5 x^2 - 4 x^4\right)\), we will follow these steps:
### Step 1: Expand Both Expressions
First, write down the expressions within the parentheses separately:
1. First expression: \(5 x^2 - 3 x^3 + 4 x^4 + 7 x + 5\)
2. Second expression: \(7 x + 5 - 3 x^3 + 5 x^2 - 4 x^4\)
### Step 2: Distribute the Negative Sign to the Second Expression
We'll distribute the negative sign over the second expression:
[tex]\[
-(7 x + 5 - 3 x^3 + 5 x^2 - 4 x^4) = -7 x - 5 + 3 x^3 - 5 x^2 + 4 x^4
\][/tex]
### Step 3: Combine Both Expressions
Now, combine the first expression and the negation of the second expression:
[tex]\[
(5 x^2 - 3 x^3 + 4 x^4 + 7 x + 5) + (-7 x - 5 + 3 x^3 - 5 x^2 + 4 x^4)
\][/tex]
### Step 4: Group Like Terms
Group together the terms with the same powers of \(x\):
[tex]\[
(5 x^2 - 5 x^2) + (-3 x^3 + 3 x^3) + (4 x^4 + 4 x^4) + (7 x - 7 x) + (5 - 5)
\][/tex]
### Step 5: Simplify Each Group
Simplify each group of like terms:
1. \(5 x^2 - 5 x^2 = 0\)
2. \(-3 x^3 + 3 x^3 = 0\)
3. \(4 x^4 + 4 x^4 = 8 x^4\)
4. \(7 x - 7 x = 0\)
5. \(5 - 5 = 0\)
### Step 6: Combine the Results
Adding up all the simplified terms:
[tex]\[
0 + 0 + 8 x^4 + 0 + 0 = 8 x^4
\][/tex]
So, the simplified expression is \(8 x^4\).
Thus, the correct answer is:
C) [tex]\(8 x^4\)[/tex]
