Sure, let's add the given polynomials step by step.
We have two polynomials:
[tex]\[ 5x^2 + 2x + 2 \][/tex]
and
[tex]\[ -6x^3 + 8x + 9 \][/tex]
To add these, we need to align the terms with the same degree. Since the first polynomial is of degree 2 and the second is of degree 3, we can write them in a way to clearly see their corresponding coefficients:
1. The polynomial [tex]\(5x^2 + 2x + 2\)[/tex] can be rewritten to include the [tex]\(x^3\)[/tex] term even though it has a coefficient of 0. It will look like this:
[tex]\[ 0x^3 + 5x^2 + 2x + 2 \][/tex]
2. The second polynomial [tex]\(-6x^3 + 8x + 9\)[/tex] already has the [tex]\(x^3\)[/tex] term:
[tex]\[ -6x^3 + 0x^2 + 8x + 9 \][/tex]
Now, we align these polynomials and add their coefficients term by term:
[tex]\[
\begin{array}{r}
0x^3 + 5x^2 + 2x + 2 \\
+ (-6x^3 + 0x^2 + 8x + 9) \\
\hline
\end{array}
\][/tex]
Let's add the coefficients for each power of [tex]\(x\)[/tex]:
- [tex]\(x^3\)[/tex] terms: [tex]\(0 + (-6) = -6\)[/tex]
- [tex]\(x^2\)[/tex] terms: [tex]\(5 + 0 = 5\)[/tex]
- [tex]\(x\)[/tex] terms: [tex]\(2 + 8 = 10\)[/tex]
- Constant terms: [tex]\(2 + 9 = 11\)[/tex]
So, the resulting polynomial after adding these together is:
[tex]\[
-6x^3 + 5x^2 + 10x + 11
\][/tex]
Thus, the sum of the polynomials [tex]\( \left(5x^2 + 2x + 2\right) + \left(-6x^3 + 8x + 9\right) \)[/tex] is [tex]\(\boxed{-6x^3 + 5x^2 + 10x + 11}\)[/tex].
