Divide the rational expression.

a^3+x^3\2a^3-2a+2x^2÷a^2-x^2\10.

Wow, I see 4 questions all typed in an un-decipherable way, showing no brackets, and showing no effort on your part.

I assume there is a division in there somewhere, but why use / in one case , \in another, and then also ÷ ??

We use / to show division.

Please re-type, using proper notation.

(a^3+x^3)/(2a^3-2ax+3x^2)÷(a^2-x^2)/(10)

To divide the rational expression, we need to follow the steps of polynomial division. Here's how you can do it:

First, let's rewrite the expression in a clearer way, so we can better understand the division:

(a^3 + x^3) / (2a^3 - 2a + 2x^2) ÷ (a^2 - x^2) / 10

For a clearer representation, let's take each polynomial term and arrange them in descending order:

(a^3 + x^3) / (2a^3 - 2a + 2x^2) ÷ (a^2 - x^2) / 10

Next, let's divide each term one by one:

1. Divide the first term of the numerator (a^3) by the first term of the denominator (a^2). The result is 'a'.

2. Multiply 'a' by the second part of the denominator (a^2 - x^2) to get (a)(a^2 - x^2) = a^3 - ax^2.

3. Subtract (a^3 - ax^2) from the original numerator (a^3 + x^3) to get x^3 - ax^2.

4. Now, bring down the next term, which is 'x^3'. We have (x^3 - ax^2).

5. Divide the first term of (x^3 - ax^2) by the first term of the denominator (a^2). The result is (x^3/a^2).

6. Multiply (x^3/a^2) by the second part of the denominator (a^2 - x^2) to get (x^3/a^2) * (a^2 - x^2) = x^3 - x^5/a.

7. Subtract (x^3 - x^5/a) from (x^3 - ax^2) to get -ax^2 + x^5/a.

Now, we have the partially divided expression: a + (x^3 - ax^2)/a^2 - (ax^2 - x^5/a)/(2a^3 - 2a + 2x^2).

To complete the division, we need to perform additional steps, but since the expression is quite complex, manual division becomes cumbersome. It's recommended to use computer software or calculators that can simplify and finalize the division process accurately.