I have obtained as the expression for some quantity the following gargantuan formula:

$$ \frac{k^8 + 3k^7 + 8k^6 + 3k^5 - 16k^4 - 32k^3 + 63k^2 - 34k + 6}{k^6 + 3k^5 + 6k^4 - 24k^2 + 21k - 5}$$.

What I really need is a (very) good lower bound on it, that will hopefully be a more manageable expression.

Is there a systematic way of finding such bounds?

**The bound needs to be valid only on $[5,\infty]$.**

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