I do not know how to set up long division on this site so I will do my best without it:
Based on the assumption you mean: (s^2+s -8) div (x+4)(s2+s−8)÷(x+4)
There is nothing to associate xx with ss so the only thing we can reasonably divide are the constants of (-8) and 4. We would then be left over with some form of unsolvable fraction which would have to be written 'as is'.
Write (s^2+s -8) div (x+4)" as "(s^2+s -8)/(x+4)(s2+s−8)÷(x+4) as s2+s−8x+4
considering just the constants: (-8) div 4 = -2(−8)÷4=−2
So the first part of our solution is -2
If the division gives -2 then we have to subtract
-2 times (x+4) = -2x - 8−2×(x+4)=−2x−8 ................ ( 1 )
from the original expression to determine what is left. This in turn will also have to be divided by ( x + 4 )(x+4). This process is like the old remainder system you would have been taught some years back.
so we have in effect a remainder of:
( (s^2 + s -8 ) - (-2)(x+4))/(x+4)(s2+s−8)−(−2)(x+4)x+4
( (s^2 + s -8 ) +2(x+4))/(x+4)(s2+s−8)+2(x+4)x+4 ....... ( 2 )
Adding the remainder to the first part we have:
"part (1) + part(2)"-> -2 + ( (s^2 + s -8 ) +2(x+4))/(x+4)part (1) + part(2)→−2+(s2+s−8)+2(x+4)x+4
This would give:
-2 + (s^2+s +2x)/(x+4)−2+s2+s+2xx+4
or -2 + (s^2+s)/(x+4) + (2x)/(x+4)−2+s2+sx+4+2xx+4
Addendum: Checking the solution
If my solution is correct then multiplying it by (x+4)(x+4) should give us the original expression. This would take the form:
(x+4)[-2+(s^2+s+2x)/(x+4)](x+4)[−2+s2+s+2xx+4]
=> -2x -8 +s^2 + s +2x⇒−2x−8+s2+s+2x
which gives us: s^2 + s -8s2+s−8
so correct!!!!
