Let's add the polynomials [tex]\(\left(g^2 - 4g^4 + 5g + 9\right) + \left(-3g^3 + 3g^2 - 6\right)\)[/tex] step by step:
1. Rewrite terms that are subtracted as addition of the opposite:
[tex]\[
g^2 + \left(-4g^4\right) + 5g + 9 + \left(-3g^3\right) + 3g^2 + (-6)
\][/tex]
2. Group like terms:
Combine the terms with the same degrees of [tex]\(g\)[/tex]:
- Terms with [tex]\(g^4\)[/tex]: [tex]\(-4g^4\)[/tex]
- Terms with [tex]\(g^3\)[/tex]: [tex]\(-3g^3\)[/tex]
- Terms with [tex]\(g^2\)[/tex]: [tex]\(g^2 + 3g^2\)[/tex]
- Terms with [tex]\(g\)[/tex]: [tex]\(5g\)[/tex]
- Constant terms: [tex]\(9 + (-6)\)[/tex]
3. Combine like terms:
- Combine the [tex]\(g^2\)[/tex] terms: [tex]\(g^2 + 3g^2 = 4g^2\)[/tex]
- Combine the constant terms: [tex]\(9 + (-6) = 3\)[/tex]
Now list all terms:
- [tex]\(-4g^4\)[/tex]
- [tex]\(-3g^3\)[/tex]
- [tex]\(4g^2\)[/tex]
- [tex]\(5g\)[/tex]
- [tex]\(3\)[/tex]
4. Write the resulting polynomial in standard form:
Arrange from the highest power of [tex]\(g\)[/tex] to the lowest power:
[tex]\[
-4g^4 - 3g^3 + 4g^2 + 5g + 3
\][/tex]
Therefore, the sum of the polynomials [tex]\(\left(g^2 - 4g^4 + 5g + 9\right) + \left(-3g^3 + 3g^2 - 6\right)\)[/tex] is:
[tex]\[
\boxed{-4g^4 - 3g^3 + 4g^2 + 5g + 3}
\][/tex]
