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\begin{align*}
R_{n}(x) &=\frac{1}{n!} \int_{a}^{b} f^{n+1} (t)(x-t)_{+}^ndt \\
&= \frac{1}{n!} \int_{a}^{x} f^{n+1} (t)(x-t)_{+}^ndt +  \int_{x}^{b} f^{n+1} (t)(x-t)_{+}^ndt \\
&= \dfrac{f^{n+1}(\eta)}{n!} \int_{a}^{x}(x-t)^n dt \\
&= \dfrac{f^{n+1}(\eta)}{n!} \Big[\dfrac{-(x-t)^{n+1}}{n+1} \Big]_{a}^x =  \dfrac{f^{n+1}(\eta)}{(n+1)!} (x-a)^{n+1} , a<\eta <x 
\end{align*}
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