To find the sum of the polynomials [tex]\(-x^2 + 9\)[/tex] and [tex]\(-3x^2 - 11x + 4\)[/tex], we need to add the corresponding coefficients of the terms with the same degree (i.e., the constant terms, the linear terms, and the quadratic terms). Let's break it down step by step.
1. Identify the coefficients of each term in the first polynomial [tex]\(-x^2 + 9\)[/tex]:
- The coefficient of [tex]\(x^2\)[/tex] is [tex]\(-1\)[/tex].
- The coefficient of [tex]\(x\)[/tex] is [tex]\(0\)[/tex] (since there is no [tex]\(x\)[/tex] term).
- The constant term is [tex]\(9\)[/tex].
2. Identify the coefficients for the second polynomial [tex]\(-3x^2 - 11x + 4\)[/tex]:
- The coefficient of [tex]\(x^2\)[/tex] is [tex]\(-3\)[/tex].
- The coefficient of [tex]\(x\)[/tex] is [tex]\(-11\)[/tex].
- The constant term is [tex]\(4\)[/tex].
3. Add the coefficients of the corresponding terms from the two polynomials:
- For the [tex]\(x^2\)[/tex] terms: [tex]\(-1 + (-3)\ = -4\)[/tex].
- For the [tex]\(x\)[/tex] terms: [tex]\(0 + (-11)\ = -11\)[/tex].
- For the constant terms: [tex]\(9 + 4\ = 13\)[/tex].
4. Combine these sums to form the resulting polynomial:
[tex]\[
-4x^2 - 11x + 13
\][/tex]
Therefore, the sum of the polynomials [tex]\(-x^2 + 9\)[/tex] and [tex]\(-3x^2 - 11x + 4\)[/tex] is:
[tex]\[
\boxed{-4x^2 - 11x + 13}
\][/tex]
