Let's start by adding the polynomials step by step.
First, consider the two polynomials:
[tex]\[ (6x + 7 + x^2) \][/tex]
and
[tex]\[ (2x^2 - 3) \][/tex]
When we add these two polynomials, we combine the like terms (terms with [tex]\(x^2\)[/tex], terms with [tex]\(x\)[/tex], and constant terms).
Combining the like terms, we get:
[tex]\[
x^2 + 2x^2 = 3x^2
\][/tex]
[tex]\[
6x = 6x \quad \text{(since second polynomial has no x term)}
\][/tex]
[tex]\[
7 + (-3) = 4
\][/tex]
So, the sum of the first two polynomials is:
[tex]\[
3x^2 + 6x + 4
\][/tex]
Next, we need to add the polynomial:
[tex]\[
-x^2 + 6x + 4
\][/tex]
to the result of the previous addition:
[tex]\[
3x^2 + 6x + 4
\][/tex]
Combining these terms, we get:
[tex]\[
3x^2 + (-x^2) = 2x^2
\][/tex]
[tex]\[
6x + 6x = 12x
\][/tex]
[tex]\[
4 + 4 = 8
\][/tex]
Thus, the resulting polynomial is:
[tex]\[
2x^2 + 12x + 8
\][/tex]
As such, the sum of the polynomials is:
[tex]\[ \boxed{2x^2 + 12x + 8} \][/tex]
