Let's go step-by-step to add the polynomial expressions [tex]\((g^2 - 4g^4 + 5g + 9) + (-3g^3 + 3g^2 - 6)\)[/tex]:
1. Rewrite as addition of the opposite:
[tex]\[
g^2 + (-4g^4) + 5g + 9 + (-3g^3) + 3g^2 + (-6)
\][/tex]
2. Group like terms:
[tex]\[
(-4g^4) + (-3g^3) + (g^2 + 3g^2) + 5g + (9 + (-6))
\][/tex]
3. Combine like terms:
- The [tex]\(g^4\)[/tex] term: [tex]\(-4g^4\)[/tex]
- The [tex]\(g^3\)[/tex] term: [tex]\(-3g^3\)[/tex]
- The [tex]\(g^2\)[/tex] term: [tex]\(g^2 + 3g^2 = 4g^2\)[/tex]
- The [tex]\(g\)[/tex] term: [tex]\(5g\)[/tex]
- The constant term: [tex]\(9 + (-6) = 3\)[/tex]
4. Write the resulting polynomial in standard form:
[tex]\[
-4g^4 + -3g^3 + 4g^2 + 5g + 3
\][/tex]
So the sum of the polynomial expressions is:
[tex]\[
-4g^4 + -3g^3 + 4g^2 + 5g + 3
\][/tex]
Among the given choices, the correct one is:
[tex]\[
\boxed{-4g^4 - 3g^3 + 4g^2 + 5g + 3}
\][/tex]
