To simplify the given expression:
[tex]\[
(5x^2 + 2x + 6) + (x^3 - x - 2) - (7x^3 - 7x^2 - 5)
\][/tex]
we will follow several steps:
### Step 1: Distribute the negative sign in the third term
The given expression is:
[tex]\[
(5x^2 + 2x + 6) + (x^3 - x - 2) - (7x^3 - 7x^2 - 5)
\][/tex]
First, distribute the negative sign across the terms in the third expression:
[tex]\[
7x^3 - 7x^2 - 5 \rightarrow - (7x^3) + (7x^2) + 5
\][/tex]
So, the expression now becomes:
[tex]\[
(5x^2 + 2x + 6) + (x^3 - x - 2) + (-7x^3 + 7x^2 + 5)
\][/tex]
### Step 2: Combine like terms
Let's identify and group the like terms:
Cubic terms: [tex]\(x^3\)[/tex]
[tex]\[
x^3 - 7x^3 = -6x^3
\][/tex]
Quadratic terms: [tex]\(x^2\)[/tex]
[tex]\[
5x^2 + 7x^2 = 12x^2
\][/tex]
Linear terms: [tex]\(x\)[/tex]
[tex]\[
2x - x = x
\][/tex]
Constant terms:
[tex]\[
6 - 2 + 5 = 9
\][/tex]
### Step 3: Sum up all the simplified groups
Now we can rewrite the simplified expression by summing up the results of each grouped term:
[tex]\[
-6x^3 + 12x^2 + x + 9
\][/tex]
So, the simplified form of the given expression is:
[tex]\[
\boxed{-6x^3 + 12x^2 + x + 9}
\][/tex]
