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Denote by $\Sigma n^m$ the sum of the $m$-th powers of the first $n$ positiveintegers $1^m+2^m+\ldots +n^m$. Similarly let $\Sigma^r n^m$ be the $r$-foldsum of the $m$-th powers of the first $n$ positive integers, defined such that$\Sigma^0 n^{m}=n^m$, and then recursively by $\Sigma^{r+1} n^{m}=\Sigma^{r}1^{m}+\Sigma^{r} 2^{m}+\ldots + \Sigma^{r} n^{m}$. During the early17th-century, polynomial expressions for the sums $\Sigma^r n^m$ and theirfactorisation and polynomial basis representation properties were examined byJohann Faulhaber, who published some remarkable theorems relating to these$r$-fold sums in his Academia Algebrae (1631).  In this paper we consider families of polynomials related to the Fibonacciand Lucas polynomials which naturally lend themselves to representing sums anddifferences of integer powers, as well as $\Sigma^r n^m$. Using summations overpolynomial basis representations for monomials we generalise this sum for anypolynomial $q(x)\in\mathbb{Q}[x]$, encountering some interesting coefficientfamilies in the process. Other results include using our polynomial expressionsto state some theorems of Fermat, where in particular, we obtain an explicitexpression for the quotient in Fermat's Little Theorem.  In the final two sections we examine these sums in the context of the threepairs of Stirling number varieties and binomial decompositions. We also derivean expression for $\Sigma n^m$ in terms of the Stirling numbers and Pochhammersymbols, leading to a seemingly new relation between the Stirling numbers andthe Bernoulli numbers.

On Fibonacci Polynomial Expressions for Sums of $m$th Powers, their implications for Faulhaber's Formula and some Theorems of Fermat