To subtract the given polynomials, [tex]\((2x^2 + 4x + 3)\)[/tex] and [tex]\((4x^2 - 2x - 3)\)[/tex], we need to combine like terms by subtracting the corresponding coefficients of the second polynomial from the coefficients of the first polynomial. Here’s the step-by-step process:
1. Identify coefficients for each term:
- For [tex]\(2x^2 + 4x + 3\)[/tex]:
- Coefficient of [tex]\(x^2\)[/tex]: [tex]\(2\)[/tex]
- Coefficient of [tex]\(x\)[/tex]: [tex]\(4\)[/tex]
- Constant term: [tex]\(3\)[/tex]
- For [tex]\(4x^2 - 2x - 3\)[/tex]:
- Coefficient of [tex]\(x^2\)[/tex]: [tex]\(4\)[/tex]
- Coefficient of [tex]\(x\)[/tex]: [tex]\(-2\)[/tex]
- Constant term: [tex]\(-3\)[/tex]
2. Subtract the coefficients of corresponding terms:
- Coefficient of [tex]\(x^2\)[/tex]:
[tex]\[
2 - 4 = -2
\][/tex]
- Coefficient of [tex]\(x\)[/tex]:
[tex]\[
4 - (-2) = 4 + 2 = 6
\][/tex]
- Constant term:
[tex]\[
3 - (-3) = 3 + 3 = 6
\][/tex]
3. Combine the results:
[tex]\[
(-2)x^2 + 6x + 6
\][/tex]
So, the result of subtracting the polynomials [tex]\((2x^2 + 4x + 3) - (4x^2 - 2x - 3)\)[/tex] is:
[tex]\[
-2x^2 + 6x + 6
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{D. \ -2x^2 + 6x + 6}
\][/tex]
