To add the given polynomials, we need to combine the corresponding coefficients of each term. Here are the steps:
1. Identify the polynomials:
[tex]\[
\left(-3z^2 - 9z + 6\right) \quad \text{and} \quad \left(-2z^2 + 2z + 2\right)
\][/tex]
2. Combine the coefficients of [tex]\( z^2 \)[/tex]:
[tex]\[
-3z^2 + -2z^2 = (-3 + -2)z^2 = -5z^2
\][/tex]
3. Combine the coefficients of [tex]\( z \)[/tex]:
[tex]\[
-9z + 2z = (-9 + 2)z = -7z
\][/tex]
4. Combine the constant terms:
[tex]\[
6 + 2 = 8
\][/tex]
5. Write the resulting polynomial:
Combining all these results, the polynomial we get is:
[tex]\[
-5z^2 - 7z + 8
\][/tex]
So, the result of adding the polynomials [tex]\(\left(-3z^2 - 9z + 6\right)\)[/tex] and [tex]\(\left(-2z^2 + 2z + 2\right)\)[/tex] is:
[tex]\[
-5z^2 - 7z + 8
\][/tex]
In coefficient form, this solution can be represented as:
[tex]\[
[-5, -7, 8]
\][/tex]
And in polynomial form, the final answer is:
[tex]\[
-5z^2 - 7z + 8
\][/tex]
