# Particular integral trial solution with repeated roots

For the DE $$y”+9y’+9y=x^3e^{4x}$$, I’m confused what my trial solution should be for the particular integral. I know that if the RHS is $$P_n(x)e^{\lambda x}$$ and $$m_1, m_2$$ solution of the characteristic equation, then the trial solution is normally $$x^kQ_n(x)e^{\lambda x}$$, where:

$$k=0$$ if $$m_1 \neq m_2 \neq \lambda$$

$$k=1$$ if $$m_1 \neq m_2 = \lambda$$

and $$k=2$$ if $$m_1=m_2=\lambda$$.

But my situation is the case $$m_1=m_2 \neq \lambda$$.

Intuitively I feel like I should try $$Q_{n+1}(x)e^{\lambda x}$$, but I have no idea if this is correct – or, if it is, why.