Let's simplify the given expression step-by-step:
The original expression is:
[tex]\[
(5x^2 + 2x + 6) + (x^3 - x - 2) - (7x^3 - 7x^2 - 5)
\][/tex]
First, distribute the negative sign through the third term:
[tex]\[
(5x^2 + 2x + 6) + (x^3 - x - 2) - 7x^3 + 7x^2 + 5
\][/tex]
Next, combine like terms by combining the coefficients of the same power of [tex]\(x\)[/tex]:
For [tex]\(x^3\)[/tex]:
[tex]\[
x^3 - 7x^3 = -6x^3
\][/tex]
For [tex]\(x^2\)[/tex]:
[tex]\[
5x^2 + 7x^2 = 12x^2
\][/tex]
For [tex]\(x\)[/tex]:
[tex]\[
2x - x = x
\][/tex]
For the constant terms:
[tex]\[
6 - 2 + 5 = 9
\][/tex]
Now, combine all the simplified terms:
[tex]\[
-6x^3 + 12x^2 + x + 9
\][/tex]
So, the simplified expression is:
[tex]\[
-6x^3 + 12x^2 + x + 9
\][/tex]
