The method we need to use here is very particular to this type of a situation. The way we will find that polynomial, or the divisor, is to follow this formula: [tex]g(x)= \frac{dividend-remainder}{quotient} [/tex]. For us that will look like this: [tex]g(x)= \frac{4x^4-5x^3-39x^2-46x-2-(-5x+8)}{x^2-3x-5} [/tex]. First we will simplify as much as possible that very long numerator there. It simplifies to [tex]g(x)= \frac{4x^4-5x^3-39x^2-41x-10}{x^2-3x-5} [/tex]. What you do now is use long division of polynomials, which, unfortunately, is impossible to show in this forum. However, get familiar with the long division process if you are not already, and you will find that your polynomial g(x) is [tex]4x^2+7x+2[/tex].
