Divide the rational expressions below and simplify as much as possible:

Apply fraction rule to flip the second fraction to times the two fractions...

< x^2+8 / 2x^2+x-3 * x-1 / x+2 >

Then you can multiply the fractions:
< (x^2+8)(x-1) / (2x^2+x-3)(x+2) >

Factor (2x^2+x-3) as (x-1)(2x+3) so we can cancel out (x-1) from the denominator and numerator < (x^2+8)(x-1) / (x-1)(2x+3)(x+2) >
< = x^2+8 / (2x+3)(x+2) >

Expand denominator (2x+3)(x+2) with foil method shown in image:

Therefore the answer is < = x^2+8 / 2x^2+7x+6 >