<s>[INST] About factoring and simplifying the following algebra expression... today  a problem like the following was sent to me by a friend: $$\dfrac{(a^3+a+10)}{(a^3+3a^2+a-2)}\cdot\dfrac{(a^2-2a+5)}{(a^4-3a^2+1)}$$ find the simplest form of the expression above. The answer as the key suggests turns out to be $a^2-a-1$. 
I've tried to factor out $(a+1)$ but there were too many factors. It is possible to do it using the polynomial division. But it will take too long. There is a general problem about the fact that these questions take too long. Is there any way to simplify only using the factors of the variables?
What are your suggestions?
 [/INST] Probably there is a typo. The reducible form occurs if the operation is a division, not a multiplication. Accordingly, inverting the numerator and the denominator of the second fraction, the expression becomes 
$$\dfrac{(a^3+a+10)}{(a^3+3a^2+a-2)}\cdot\dfrac{(    a^4-3a^2+1    )}{(  a^2-2a+5  )}$$
and taking into account that 
$$a^3+a+10=(a+2)(a^2-2a+5)$$
$$a^3+3a^2+a-2=(a+2)(a^2+a-1)$$
$$a^4-3a^2+1 =(a^2-a+1)(a^2+a-1)$$
we get that the whole expression reduces to $a^2-a+1\,\,$.
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