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Question

# Factor. Assume that variables used as exponents represent positive integers.$2 x^{2 n}+11 x^n+5$

Solution

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\begin{align*} 2x^{2n}+11x^{n}+5&=2(x^n)^2+11x^n+5\\ &=2y^2+11y+5 \hspace{1cm}\textcolor{#4257b2}{\textrm{Let }x^n=y}\\ &=2y^2+y+10y+5 \\ &=(2y^2+y)+5(2y+1) \\ &=y(2y+1)+5(2y+1) \\ &=(y+5)(2y+1) \\ &=(x^n+5)(2x^n+1) \\ \end{align*}

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