To find the sum of the polynomials [tex]\((8 x^2 - 9 y^2 - 4 x)\)[/tex] and [tex]\((x^2 - 3 y^2 - 7 x)\)[/tex], we must add the coefficients of like terms.
Here’s the step-by-step solution:
1. Identify the like terms:
- The [tex]\(x^2\)[/tex]-terms are [tex]\(8 x^2\)[/tex] from the first polynomial and [tex]\(x^2\)[/tex] from the second polynomial.
- The [tex]\(y^2\)[/tex]-terms are [tex]\(-9 y^2\)[/tex] from the first polynomial and [tex]\(-3 y^2\)[/tex] from the second polynomial.
- The [tex]\(x\)[/tex]-terms are [tex]\(-4 x\)[/tex] from the first polynomial and [tex]\(-7 x\)[/tex] from the second polynomial.
2. Add the coefficients of the [tex]\(x^2\)[/tex]-terms:
[tex]\[8 x^2 + x^2 = (8 + 1) x^2 = 9 x^2\][/tex]
3. Add the coefficients of the [tex]\(y^2\)[/tex]-terms:
[tex]\[-9 y^2 - 3 y^2 = (-9 - 3) y^2 = -12 y^2\][/tex]
4. Add the coefficients of the [tex]\(x\)[/tex]-terms:
[tex]\[-4 x - 7 x = (-4 - 7) x = -11 x\][/tex]
Putting it all together, the sum of the polynomials is:
[tex]\[9 x^2 - 12 y^2 - 11 x\][/tex]
Therefore, the correct answer is:
[tex]\[9 x^2 - 12 y^2 - 11 x\][/tex]
So, the sum of the polynomials is:
[tex]\[
\boxed{9 x^2 - 12 y^2 - 11 x}
\][/tex]
