To solve the problem of adding the given polynomials, we will combine like terms step by step. Here are the two polynomials we need to add:
[tex]\[
(-4x^4 + 3x^2 + 14) + (-3x^4 - 14x^2 - 8)
\][/tex]
Step 1: Write down the polynomials next to each other:
[tex]\[
-4x^4 + 3x^2 + 14 \\
-3x^4 - 14x^2 - 8
\][/tex]
Step 2: Combine the like terms. Start with the highest degree terms and move to the constant term:
### Degree 4 Terms:
Combine [tex]\(-4x^4\)[/tex] and [tex]\(-3x^4\)[/tex]:
[tex]\[
-4x^4 + (-3x^4) = -7x^4
\][/tex]
### Degree 2 Terms:
Combine [tex]\(3x^2\)[/tex] and [tex]\(-14x^2\)[/tex]:
[tex]\[
3x^2 + (-14x^2) = 3x^2 - 14x^2 = -11x^2
\][/tex]
### Constant Terms:
Combine [tex]\(14\)[/tex] and [tex]\(-8\)[/tex]:
[tex]\[
14 + (-8) = 14 - 8 = 6
\][/tex]
Step 3: Put all the combined terms together:
[tex]\[
-7x^4 - 11x^2 + 6
\][/tex]
So, the result of adding the given polynomials is:
[tex]\[
-7x^4 - 11x^2 + 6
\][/tex]
This is the simplified form of the sum of the polynomials [tex]\((-4x^4 + 3x^2 + 14)\)[/tex] and [tex]\((-3x^4 - 14x^2 - 8)\)[/tex].
