To find the sum of the polynomials \((6x + 7 + x^2) + (2x^2 - 3) - x^2 + 6x + 4\), we will combine like terms from each polynomial. Let's break this process down step-by-step.
First, write down each polynomial and align the like terms (terms involving the same power of \(x\)) together:
1. \(6x + 7 + x^2\)
2. \(2x^2 - 3\)
3. \(-x^2 + 6x + 4\)
Now, we will add these polynomials together term by term.
### Combining \(x^2\) terms:
- From the first polynomial: \(+ x^2\)
- From the second polynomial: \(+ 2x^2\)
- From the third polynomial: \(- x^2\)
Combining these, we get:
[tex]\[ x^2 + 2x^2 - x^2 = 2x^2 \][/tex]
### Combining \(x\) terms:
- From the first polynomial: \(+ 6x\)
- From the second polynomial: None (\(0x\))
- From the third polynomial: \(+ 6x\)
Combining these, we get:
[tex]\[ 6x + 6x = 12x \][/tex]
### Combining constant terms:
- From the first polynomial: \(+ 7\)
- From the second polynomial: \(- 3\)
- From the third polynomial: \(+ 4\)
Combining these, we get:
[tex]\[ 7 - 3 + 4 = 8 \][/tex]
Finally, putting all the terms together, we get the sum of the polynomials:
[tex]\[ 2x^2 + 12x + 8 \][/tex]
Therefore, the sum of the polynomials \((6x + 7 + x^2) + (2x^2 - 3) - x^2 + 6x + 4\) is:
[tex]\[
\boxed{2x^2 + 12x + 8}
\][/tex]
