Sure, let's simplify the given expression step-by-step:
The expression we need to simplify is:
[tex]\[
\frac{2x^8 + 12x^6 - 10x^3}{-2x^5}
\][/tex]
### Step 1: Distribute the denominator to each term in the numerator
We can rewrite the fraction by distributing [tex]\(-2x^5\)[/tex] to each term in the numerator:
[tex]\[
\frac{2x^8}{-2x^5} + \frac{12x^6}{-2x^5} - \frac{10x^3}{-2x^5}
\][/tex]
### Step 2: Simplify each term individually
- For the first term:
[tex]\[
\frac{2x^8}{-2x^5} = -\frac{2x^8}{2x^5} = -x^{8-5} = -x^3
\][/tex]
- For the second term:
[tex]\[
\frac{12x^6}{-2x^5} = -\frac{12x^6}{2x^5} = -6 \frac{x^6}{x^5} = -6x^{6-5} = -6x
\][/tex]
- For the third term:
[tex]\[
\frac{10x^3}{-2x^5} = -\frac{10x^3}{2x^5} = -5 \frac{x^3}{x^5} = -5 x^{3-5} = -5 x^{-2} = \frac{5}{x^2}
\][/tex]
### Step 3: Combine the simplified terms
Now, we combine all the simplified terms:
[tex]\[
-x^3 - 6x + \frac{5}{x^2}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\boxed{-x^3 - 6x + \frac{5}{x^2}}
\][/tex]
