To find the sum of the polynomials [tex]\((8x^2 - 9y^2 - 4x)\)[/tex] and [tex]\((x^2 - 3y^2 - 7x)\)[/tex], we need to add the coefficients of the corresponding terms: [tex]\(x^2\)[/tex], [tex]\(y^2\)[/tex], and [tex]\(x\)[/tex].
### Step-by-Step Solution:
1. Identify the coefficients of [tex]\(x^2\)[/tex] terms in both polynomials:
- For the first polynomial: [tex]\(8x^2\)[/tex]
- For the second polynomial: [tex]\(x^2\)[/tex]
Add these coefficients together:
[tex]\[
8 + 1 = 9
\][/tex]
Hence, the coefficient of [tex]\(x^2\)[/tex] in the resulting polynomial is [tex]\(9\)[/tex].
2. Identify the coefficients of [tex]\(y^2\)[/tex] terms in both polynomials:
- For the first polynomial: [tex]\(-9y^2\)[/tex]
- For the second polynomial: [tex]\(-3y^2\)[/tex]
Add these coefficients together:
[tex]\[
-9 + (-3) = -12
\][/tex]
Hence, the coefficient of [tex]\(y^2\)[/tex] in the resulting polynomial is [tex]\(-12\)[/tex].
3. Identify the coefficients of [tex]\(x\)[/tex] terms in both polynomials:
- For the first polynomial: [tex]\(-4x\)[/tex]
- For the second polynomial: [tex]\(-7x\)[/tex]
Add these coefficients together:
[tex]\[
-4 + (-7) = -11
\][/tex]
Hence, the coefficient of [tex]\(x\)[/tex] in the resulting polynomial is [tex]\(-11\)[/tex].
### Form the Resulting Polynomial:
Combine the results from each step to form the sum of the given polynomials:
[tex]\[
9x^2 - 12y^2 - 11x
\][/tex]
### Final Answer:
The sum of the polynomials [tex]\((8x^2 - 9y^2 - 4x)\)[/tex] and [tex]\((x^2 - 3y^2 - 7x)\)[/tex] is:
[tex]\[
\boxed{9x^2 - 12y^2 - 11x}
\][/tex]
