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Consider the characteristic polynomial
|A-pI|
= (2-p)^2*(3-p)+2+2- 2(2-p) - 2(2-p) - (3-p)
= -p^3+7p^2-11p+5

So -A^3+7A^2-11A+5I = 0

Also, consider another polynomial (p^5-8p^4+19a^3-2a^2+18a-6)
by long division(or other possible means), we obtain
(p^5-8p^4+19a^3-2a^2+18a-6)
= (-p^2+p-1)(-p^3+7p^2-11p+5) + 7p^2+3p-1

So
A^5-8A^4+19A^3-2A^2+18A-6I
= (-A^2+A-I)(-A^3+7A^2-11A+5I) + 7A^2+3A-I
=  7A^2+3A - I

(The rest is left to you since multiplication of matrices is quite easy, and I don't want to type out those matrices, most importantly xD)

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