You can do long division - but let's adjust the signs first, because it is not comfortable with the divisor starting with a negative sign
#rarr" "# dividing by a negative changes the signs.
Take out a factor of #-1# in the numerator and denominator:
#(-3x^4+3x^2+2x+15)/(-x^3-x)#
#= (-(3x^4-3x^2-2x-15))/(-(x^3+x))" "larr (-/-) =+#
#= (3x^4-3x^2-2x-15)/(x^3+x)#
Leave a space for the missing term in #x^3#
#color(white)(mmmmmmmm.mmmm)3x" "larr (3x^4 div x^3 = 3x)#
#x^3 +x |bar(3x^4" "-3x^2-2x-15)#
Multiply #3x# by both terms in the divisor:
#color(white)(mmmmmmmm.mmmm)3x#
#x^3 +x |bar(3x^4" "-3x^2-2x-15)#
#color(white)(mmm.m)ul(3x^4" "" "3x^2)color(white)(xxxxxxxxx)larr# subtract
#color(white)(mmmmmm.m)-6x^2-2x" "larr# bring down next term
#(3x^4-3x^2-2x-15) div(x^3+x)#
# = 3x " remainder "-6x^2 -2x#
This can be written as:
#3x + (-6x^2-2x)/(x^3+x)#
#3x +(-2x(x+1))/(x^2(x+1))" "larr# factorise top and bottom
#= 3x -2/x#
