How to factorise $x^4 - 3x^3 + 2$, so as to compute the limit of a quotient? Question:

Find the limit: $$\lim_{x \to 1}\frac{x^4 - 3x^3 + 2}{x^3 -5x^2+3x+1}$$

The denominator can be simplified to: $$(x-1)(x^2+x)$$
However, I am unable to factor the numerator in a proper manner (so that $(x-1)$ will cancel out)
I know upon graphing that the limit is $5\over4$. What should I do here?
Note: To be done without the use of L'Hospital Rule
 A: You may use Horner here. Since $x=1$ is a root of the nominator (you can check that easily) then you can deduce that down by applying a long division.
Hence your limit is deduced down to:
$$\lim_{x\rightarrow 1}\frac{x^4-3x^3+2}{x^3-5x^2+3x+1}=\lim_{x\rightarrow 1}\frac{(x-1)\left ( x^3-2x^2-2x-2 \right )}{\left ( x-1 \right )\left ( x^2-4x-1 \right )}=\frac{1-2-2-2}{1-4-1}=5/4$$
A: Here are the steps
$$\lim_{x \to 1}\frac{x^4 - 3x^3 + 2}{x^3 -5x^2+3x+1}$$
$$= \lim_{x \to 1}\frac{(x-1)(x^3- 2x^2-2x-2)}{(x-1)(x^2-4x-1)} $$
$$= \lim_{x \to 1}\frac{x^3- 2x^2-2x-2}{x^2-4x-1} $$
$$= \frac{1- 2-2-2}{1-4-1} = \frac{1- 6}{1-5}  = \frac{5}{4} $$
A: Using the Euclidean division we get
$$\begin{array}\\x^4-3x^3+2&\Bigg|&x-1\\ 
-(x^4-x^3)&\Bigg|&x^3\\
=-2x^3+2&\Bigg|&-2x^2\\
-(-2x^3+2x^2)\\
=-2x^2+2&\Bigg|&-2x\\-(-2x^2+2x)\\=-2x+2&\Bigg|&-2
\end{array}$$
so we find that
$$x^4-3x^3+2=(x-1)(x^3-2x^2-2x-2)$$
A: Hint: The numerator can be factorized as $$x^4-3x^3+2=x^4-1-3(x^3-1)=(x-1)((x^2+1)(x+1)-3(x^2+x+1))$$ and the denominator as $$x^4-5x^2+3x+1\\=x^4-1-(5x^2-3x-2)=(x-1)((x^2+1)(x+1)-(5x+2))$$
