Find a Trial Solution for the Method of Undetermined Coefficients $y ^ { \prime \prime } + 2 y ^ { \prime } = 1 + x e ^ { - 4 x }$

Multiple Choice

Find a trial solution for the method of undetermined coefficients. Do not determine the coefficients. $y ^ { \prime \prime } + 2 y ^ { \prime } = 1 + x e ^ { - 4 x }$

A) $\begin{array} { l } y _ { p 1 } ( x ) = A \\y _ { p 2 } ( x ) = B x e ^ { 4 x }\end{array}$
B) $\begin{array} { l } y _ { p 1 } ( x ) = A x \\y _ { p 2 } ( x ) = x ( A x + B ) e ^ { - 4 x }\end{array}$
C) $\begin{array} { l } y _ { 71 } ( x ) = A \\y _ { p 2 } ( x ) = x ( A + B ) e ^ { - 4 x }\end{array}$
D) $\begin{array} { l } y _ { p 1 } ( x ) = A x \\y _ { p 2 } ( x ) = x ^ { 2 } ( A x + B ) e ^ { - 4 x }\end{array}$
E) $\begin{array} { l } y _ { p 1 } ( x ) = A \\y _ { p 2 } ( x ) = x ( A + B x ) e ^ { - 4 x }\end{array}$