Given: (x^4+2x^3+5x-3)/(x+2)
Check if the numerator is divisible by x+2 by plugging x=-2 into it: (-2)^4+2(-2)^3+5(-2)-3=-13
So, the numerator is not divisible by x+2. The remainder is -13.
Factor the numerator with x+2 as a factor separating the first two terms and the second two:
x^4+2x^3+5x-3=x^3(x+2)+5(x+2)-13
Thus, (x^4+2x^3+5x-3)/(x+2)={x^3(x+2)+5(x+2)-13}/(x+2)
=x^3(x+2)/(x+2)+5(x+2)/(x+2) -13/(x+2)=x^3+5 -13/(x+2) Here, -13 is not divisible by x+2.
Therefore, the answer is: (x^4+2x^3+5x-3)/(x+2)=x^3+5 R(-13)