To determine which of the given expressions is equivalent to [tex]\(\left(12x^4 - 4x^3 + 7x + 6\right) + \left(3x^2 - 4x + 14\right)\)[/tex], we will follow these steps to add the two polynomials together:
1. Write both polynomials in standard form, aligning like terms:
[tex]\[
12x^4 - 4x^3 + 0x^2 + 7x + 6
\][/tex]
[tex]\[
0x^4 + 0x^3 + 3x^2 - 4x + 14
\][/tex]
2. Add the corresponding coefficients of the like terms:
- Combine the [tex]\(x^4\)[/tex] terms: [tex]\(12x^4 + 0x^4 = 12x^4\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(-4x^3 + 0x^3 = -4x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(0x^2 + 3x^2 = 3x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(7x - 4x = 3x\)[/tex]
- Combine the constant terms: [tex]\(6 + 14 = 20\)[/tex]
3. Write down the resulting polynomial:
[tex]\[
12x^4 - 4x^3 + 3x^2 + 3x + 20
\][/tex]
After following these steps, we find that the polynomial expression equivalent to [tex]\(\left(12x^4 - 4x^3 + 7x + 6\right) + \left(3x^2 - 4x + 14\right)\)[/tex] is:
[tex]\[
12x^4 - 4x^3 + 3x^2 + 3x + 20
\][/tex]
Thus, the answer is:
[tex]\[
\boxed{A}
\][/tex]
