To check the student's work, we need to look at each step of their polynomial subtraction and see if they have performed it correctly.
The original expression to simplify is:
\[
(3x^4 - 8x^2 + 4x) - (9x - 5x^4 + 2x^2)
\]
### Step 1:
The student rewrote the expression as:
\[
3x^4 - 8x^2 + 4x - 9x + 5x^4 + 2x^2
\]
This step is correct. They correctly distributed the negative sign across the second polynomial.
### Step 2:
Next, the student combined like terms:
- From \(3x^4\) and \(5x^4\): \(3x^4 + 5x^4 = 8x^4\)
- From \(-8x^2\) and \(2x^2\): \(-8x^2 + 2x^2 = -6x^2\)
- From \(4x\) and \(-9x\): \(4x - 9x = -5x\)
So, the expression becomes:
\[
8x^4 - 6x^2 - 5x
\]
This step is also correct.
### Final Answer:
The final simplified expression given by the student is:
\[
8x^4 - 6x^2 - 5x
\]
### Conclusion:
The student is correct in their work. There were no errors in their steps. The final answer is indeed:
\[
8x^4 - 6x^2 - 5x
\]
Their friend should inform them that they did an excellent job with the polynomial subtraction.