To determine which polynomial represents the difference [tex]\((7x^2 + 8) - (4x^2 + x + 6)\)[/tex], we need to follow these steps:
1. Rewrite the expression for clarity:
[tex]\[
(7x^2 + 8) - (4x^2 + x + 6)
\][/tex]
2. Distribute the negative sign through the second polynomial:
[tex]\[
7x^2 + 8 - 4x^2 - x - 6
\][/tex]
3. Combine like terms:
Group the [tex]\(x^2\)[/tex] terms: [tex]\(7x^2 - 4x^2\)[/tex]
Group the [tex]\(x\)[/tex] terms: [tex]\(-x\)[/tex] (no other [tex]\(x\)[/tex] term to combine with)
Group the constant terms: [tex]\(8 - 6\)[/tex]
4. Simplify the grouped terms:
Combine the [tex]\(x^2\)[/tex] terms: [tex]\(7x^2 - 4x^2 = 3x^2\)[/tex]
Keep the [tex]\(x\)[/tex] term as is: [tex]\(-x\)[/tex]
Combine the constants: [tex]\(8 - 6 = 2\)[/tex]
5. Write the simplified polynomial:
[tex]\[
3x^2 - x + 2
\][/tex]
Therefore, the polynomial representing the difference is:
[tex]\[
3x^2 - x + 2
\][/tex]
Matching this with the given choices, the correct answer is:
[tex]\[
\boxed{D. \, 3x^2 - x + 2}
\][/tex]
