To add the polynomials [tex]\(\left(g^2 - 4g^4 + 5g + 9\right) + \left(-3g^3 + 3g^2 - 6\right)\)[/tex], let's follow the steps you provided:
1. Rewrite terms that are subtracted as the addition of the opposite type:
[tex]\[
g^2 + (-4g^4) + 5g + 9 + (-3g^3) + 3g^2 + (-6)
\][/tex]
2. Group like terms:
Let's group terms with the same degrees together:
[tex]\[
(-4g^4) + (-3g^3) + (g^2 + 3g^2) + 5g + (9 + -6)
\][/tex]
3. Combine like terms:
- Combine the [tex]\(g^4\)[/tex] terms:
[tex]\[
-4g^4
\][/tex]
- Combine the [tex]\(g^3\)[/tex] terms:
[tex]\[
-3g^3
\][/tex]
- Combine the [tex]\(g^2\)[/tex] terms:
[tex]\[
g^2 + 3g^2 = 4g^2
\][/tex]
- Combine the [tex]\(g\)[/tex] terms:
[tex]\[
5g
\][/tex]
- Combine the constant terms:
[tex]\[
9 + (-6) = 3
\][/tex]
4. Write the resulting polynomial in standard form:
Standard form dictates that we should write the polynomial in descending order of the degrees of [tex]\(g\)[/tex]:
[tex]\[
-4g^4 - 3g^3 + 4g^2 + 5g + 3
\][/tex]
Thus, the sum of the polynomials is:
[tex]\[
-4g^4 - 3g^3 + 4g^2 + 5g + 3
\][/tex]
From the options given, this matches the second option:
[tex]\[
-4g^4 - 3g^3 + 4g^2 + 5g + 3
\][/tex]
