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?

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 $a^2-a+1\,\,$.
