To solve the problem of adding the two polynomials [tex]\((3x^4 - 2x^3 + 1)\)[/tex] and [tex]\((12x^4 + x^2 - 11)\)[/tex], we will add the corresponding coefficients of like terms.
Here are the polynomials split into their individual terms:
1. The first polynomial is [tex]\(3x^4 - 2x^3 + 1\)[/tex]. This can be expressed in its expanded form with missing terms having coefficients of 0 for clarity:
[tex]\[
3x^4 - 2x^3 + 0x^2 + 0x + 1
\][/tex]
2. The second polynomial is [tex]\(12x^4 + x^2 - 11\)[/tex]. This can also be expanded as:
[tex]\[
12x^4 + 0x^3 + x^2 + 0x - 11
\][/tex]
Now, let's align the terms by their degrees and add the corresponding coefficients:
[tex]\[
\begin{array}{rcl}
(3x^4 + 12x^4) & : & (3 + 12)x^4 = 15x^4 \\
(-2x^3 + 0x^3) & : & (-2 + 0)x^3 = -2x^3 \\
(0x^2 + x^2) & : & (0 + 1)x^2 = 1x^2 \\
(0x + 0x) & : & (0 + 0)x = 0x \\
(1 - 11) & : & (1 - 11) = -10 \\
\end{array}
\][/tex]
Combining these results, we get the final polynomial:
[tex]\[
15x^4 - 2x^3 + x^2 + 0x - 10
\][/tex]
Simplifying, we drop the [tex]\(0x\)[/tex] term since it does not affect the polynomial:
[tex]\[
15x^4 - 2x^3 + x^2 - 10
\][/tex]
Therefore, the result of adding the given polynomials is:
[tex]\[
\boxed{15x^4 - 2x^3 + x^2 - 10}
\][/tex]
