Remainder Theorem with polynomial as divisor I'm aware that in remainder theorem you take the divisor and make it equal to zero. For the result of that just plug in x into whatever the polynomial dividend is.
But its different when the divisor is also a polynomial since it has multiple roots:
$$p(x) = \frac{2x^6 - x^4 - 4x^2 - 8}{x^2 + 4x - 3}$$

$x^2+4x-3$ into $2x^6-x^4-4x^2-8$. 
First, look at leading terms. $x^2$ into $2x^6$. That's $2x^4$. Multiply divisor ($x^2+4x-3$) by $2x^4$ (getting $2x^6+8x^5-6x^4$) and subtract from dividend ($2x^6-x^4-4x^2-8$), getting $-8x^5+5x^4-4x^2-8$. Leading terms again: $x^2$ into $-8x^5$ is $-8x^3$. Multiply divisor by $-8x^3$, subtract from $-8x^5+5x^4-4x^2-8$. Write the whole thing out the way you would write out $123$ into $456789$ and you'll see we're just doing division but with polynomials instead of numbers. Keep going until what's left of the dividend has degree less than that of the divisor, that is, less than 2. At that point what's left is the remainder, and the quotient is $2x^4-8x^3+$ whatever other terms you have worked out along the way. 
