Reciprocated Insane Roots

Algebra Level 5

Let $$a,b,c,d,e$$ be the roots to the polynomial $$f(x) = x^5 + 9x^4 + 7x^3 + 16x^2 + 6x + 7$$

The value of $$\frac {1}{a^2} + \frac {1}{b^2} + \frac {1}{c^2} + \frac {1}{d^2} + \frac {1}{e^2}$$ can be written in the form of $$- \frac {p}{q}$$ where $$p,q$$ are coprime positive integers.

What is the value of $$p-q$$?

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