Quintic Expression

Starting withn the identity $\begin{aligned} & a^5(b-c)(b-d)(c-d) - b^5(a-c)(a-d)(c-d) + c^5(a-b)(a-d)(b-d) - d^5(a-b)(a-c)(b-c) \\& = (a-b)(a-c)(a-d)(b-c)(b-d)(c-d)[a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+cd] \end{aligned}$

and then dividing by $(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)$ for distinct $a$ , $b$ , $c$ and $d$

$\frac{a^5}{(a-b)(a-c)(a-d)}+\frac{b^5}{(b-a)(b-c)(b-d)}+\frac{c^5}{(c-a)(c-b)(c-d)}+\frac{d^5}{(d-a)(d-b)(d-c)}$ $\begin{aligned} & = a^2 + b^2 + c^2 + d^2 + ab + ac + ad + bc + bd + cd\\ & = a^2 + b^2 + c^2 + d^2 + [(a + b + c + d)^2 - (a^2 + b^2 + c^2 + d^2)]/2\\ & = [a^2 + b^2 + c^2 + d^2 + (a + b + c + d)^2]/2\\ & = [45 + 3^2]/2\\ & = \boxed{27} \end{aligned}$