To perform the indicated operations on the polynomials, we will first rewrite the second polynomial \( (2^2 - 3x + 5) \) correctly and then combine all the terms.
1. Rewrite the polynomials:
- First polynomial: \( 5x^2 - 8x - 1 \)
- Second polynomial: \( 2^2 - 3x + 5 \) simplifies to \( 4 - 3x + 5 \), which can be rewritten as \( -3x + 9 \).
- Third polynomial: \( -(x^2 - 5x - 3) \) simplifies to \( -x^2 + 5x + 3 \).
Now, we combine the three polynomials:
\[
(5x^2 - 8x - 1) + (-3x + 9) + (-x^2 + 5x + 3)
\]
2. Combine all the like terms:
- \( 5x^2 - x^2 = 4x^2 \)
- \( -8x - 3x + 5x = -6x \)
- \( -1 + 9 + 3 = 11 \)
So, the resulting polynomial is:
\[
4x^2 - 6x + 11
\]
3. Write it in standard form:
The polynomial in standard form is:
\[
4x^2 - 6x + 11
\]
4. Determine the degree:
The degree of the polynomial is the highest power of \( x \) present, which in this case is:
\[
\text{Degree} = 2
\]
Thus, the final answer is:
- The polynomial: \( 4x^2 - 6x + 11 \)
- The degree: \( 2 \)