To add the given polynomials [tex]\( \left(g^2 - 4g^4 + 5g + 9\right) \)[/tex] and [tex]\( \left(-3g^3 + 3g^2 - 6\right) \)[/tex], we follow these steps:
### Step 1: Rewrite terms that are subtracted as addition of the opposite
Given the polynomials:
[tex]\[
(g^2 - 4g^4 + 5g + 9) + (-3g^3 + 3g^2 - 6)
\][/tex]
We can rewrite the expression to clearly show each term:
[tex]\[
g^2 + (-4g^4) + 5g + 9 + (-3g^3) + 3g^2 + (-6)
\][/tex]
### Step 2: Group like terms
Group the terms according to their powers of [tex]\( g \)[/tex]:
[tex]\[
(-4g^4) + (-3g^3) + (g^2 + 3g^2) + 5g + (9 - 6)
\][/tex]
### Step 3: Combine like terms
Combine the coefficients of like terms:
[tex]\[
-4g^4 - 3g^3 + (1g^2 + 3g^2) + 5g + (9 - 6)
\][/tex]
Simplifying the coefficients, we get:
[tex]\[
-4g^4 - 3g^3 + 4g^2 + 5g + 3
\][/tex]
### Step 4: Write the resulting polynomial in standard form
Combine all the terms to write the polynomial in standard form:
[tex]\[
-4g^4 - 3g^3 + 4g^2 + 5g + 3
\][/tex]
Therefore, the sum of the given polynomials is:
[tex]\[
\boxed{-4g^4 - 3g^3 + 4g^2 + 5g + 3}
\][/tex]
