Sure, let's simplify the given expression step by step:
We start with the given expression:
[tex]\[
\left(x^2 + 6x^4 - 4x^3\right) - \left(7x^4 - 4 + 8x^3\right)
\][/tex]
First, distribute the negative sign across the second expression inside the parentheses:
[tex]\[
x^2 + 6x^4 - 4x^3 - 7x^4 + 4 - 8x^3
\][/tex]
Next, combine the like terms. Let's organize the terms by their degree for clarity:
1. The [tex]\(x^4\)[/tex] terms: [tex]\(6x^4 - 7x^4 = -x^4\)[/tex]
2. The [tex]\(x^3\)[/tex] terms: [tex]\(-4x^3 - 8x^3 = -12x^3\)[/tex]
3. The [tex]\(x^2\)[/tex] term: [tex]\(x^2\)[/tex]
4. The constant term: [tex]\(4\)[/tex]
Putting it all together, we obtain:
[tex]\[
-x^4 - 12x^3 + x^2 + 4
\][/tex]
Thus, the simplified expression is:
[tex]\[
-x^4 - 12x^3 + x^2 + 4
\][/tex]
