Let's solve the problem step-by-step.
We need to subtract the polynomial [tex]\(\left(-2 f^2 - f + 3\right)\)[/tex] from the polynomial [tex]\(\left(6 f^2 - 9 f + 10\right)\)[/tex].
1. Identify the coefficients of each term:
For the first polynomial, [tex]\(\left(6 f^2 - 9 f + 10\right)\)[/tex]:
- The coefficient for [tex]\(f^2\)[/tex] is [tex]\(6\)[/tex].
- The coefficient for [tex]\(f\)[/tex] is [tex]\(-9\)[/tex].
- The constant term is [tex]\(10\)[/tex].
For the second polynomial, [tex]\(\left(-2 f^2 - f + 3\right)\)[/tex]:
- The coefficient for [tex]\(f^2\)[/tex] is [tex]\(-2\)[/tex].
- The coefficient for [tex]\(f\)[/tex] is [tex]\(-1\)[/tex].
- The constant term is [tex]\(3\)[/tex].
2. Subtract corresponding coefficients:
- For the [tex]\(f^2\)[/tex] term:
[tex]\[
6 f^2 - (-2 f^2) = 6 f^2 + 2 f^2 = 8 f^2
\][/tex]
- For the [tex]\(f\)[/tex] term:
[tex]\[
-9 f - (-f) = -9 f + f = -8 f
\][/tex]
- For the constant term:
[tex]\[
10 - 3 = 7
\][/tex]
3. Combine these results to get the final polynomial:
[tex]\[
8 f^2 - 8 f + 7
\][/tex]
So, the resulting polynomial after subtracting [tex]\(\left(-2 f^2 - f + 3\right)\)[/tex] from [tex]\(\left(6 f^2 - 9 f + 10\right)\)[/tex] is:
[tex]\[8 f^2 - 8 f + 7\][/tex]
Thus, the correct choice from the given options is:
[tex]\[8 f^2 - 8 f + 7\][/tex]
