Find a particular solution to the differential equation using the Method of Undetermined Coefficients.

x prime prime left parenthesis t right parenthesis minus $16$ x prime left parenthesis t right parenthesis plus $64$ x left parenthesis t right parenthesisx$($t$)16$x$($t$)+64$x$($t$)$equals$=4$ t e Superscript $8$ t$4$te$8$t

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Part $1$

What is the auxiliary equation associated with the given differential$$ equation?

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$($Type an equation using r as the$$ variable.$)$