<s>[INST] Algebra Problem: Division Can someone help me with a problem involving the expression $$\frac{(2x^3-3x^2+b)}{(4-x^2)}?$$
The question is to find which values $b$ can be to simplify the expression, but I do not know how to begin.
Thanks for any help you can give me.
 [/INST] $4-x^2=-(x-2)(x+2)$, so you are looking for values of $b$ that make $2x^3-3x^2+b$ divisible by either $(x-2)$ or $(x+2)$.
So if you substitute $x=2$ into $2x^3-3x^2+b$ it will be 0, and the same for $x=-2$. This give you two simple equations in $b$, which gives you two values for $b$. 
For $x=2$, you get $b=-4$, so you would expect $2x^3-3x^2-4$ to be divisible by $x-2$. You can check this by doing the division:
$2x^3-3x^2-4=(x-2)(2x^2+x+2)$.
So the expression simplifies to
$-\frac{2x^2+x+2}{x+2}$
Try this with $2x^3-3x^2+28$ and $(x+2)$, and you'll see that it works out. 
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