Chapter 8 - Linear Differential Equations of Order n - 8.3 The Method of Undetermined Coefficients: Annihilators - Problems - Page 525: 6

$F(x)=G(x)+3 H(x)$ Therefore, $(D^2-2r+5)(D^2+4)$ is the annihilator of $F(x)$.

Since, $D^2-2r+5$ is the annihilator of $G(x)= e^{x} \sin 2x$ Also, we have: $D^2+4$ is the annihilator of $H(x)=\cos 2x$.This implies that the $D^2+4$ is the annihilator of $3 H(x)=3\cos 2x$ So, $F(x)=G(x)+3 H(x)$ Therefore, $(D^2-2r+5)(D^2+4)$ is the annihilator of $F(x)$.