simplify the rational expression x^4+3x^3-2x^2-6x/x^3-2x stating all restrictions on the variable.

factor the top by grouping

x^3(x+3) - 2x(x+3)
= (x+3)(x^3 - 2x)

so wouldn't the last part cancel

so you end up with x+3

for the restriction, your divisor cannot be zero, i.e.
x^3-2x ≠ 0
x(x^2 - 2) ≠ 0
x ≠ 2 , ±√2

To simplify the rational expression (x^4 + 3x^3 - 2x^2 - 6x) / (x^3 - 2x), we need to factor both the numerator and the denominator and look for any common factors that can be canceled out.

Step 1: Factor the numerator (x^4 + 3x^3 - 2x^2 - 6x) by grouping:

= (x^4 + 3x^3) - (2x^2 + 6x)
= x^3(x + 3) - 2x(x + 3)
= (x^3 - 2x)(x + 3)

Step 2: Factor the denominator (x^3 - 2x) by factoring out the greatest common factor, which is x:

= x(x^2 - 2)
= x(x + √2)(x - √2) (using difference of squares)

Now, we can rewrite the rational expression as:

(x^3 - 2x)(x + 3) / x(x + √2)(x - √2)

To find the restrictions on the variable, we need to consider the factors in the denominator. Any value of x that makes any of these factors equal to zero will result in an undefined expression. Therefore, the restrictions on the variable are:

1. x ≠ 0 (since it would make the whole expression undefined because of the x factor in the denominator).
2. x ≠ -√2 (since it would make the expression undefined due to the (x + √2) factor in the denominator).
3. x ≠ √2 (since it would make the expression undefined due to the (x - √2) factor in the denominator).

Thus, the simplified rational expression is (x^3 - 2x)(x + 3) / x(x + √2)(x - √2), with the restrictions x ≠ 0, x ≠ -√2, and x ≠ √2.