= Solution
Write the <linear differential equation> as $\dot{\mathbf x}=M\mathbf x$, with
$$
M=\begin{pmatrix}-4&-3\\3&-4\end{pmatrix}.
$$
Its <characteristic polynomial> is $(\lambda+4)^2+9$. Thus a choice of <eigenvalues> and <eigenvectors> is
$$
\boxed{\lambda_1=-4+3i,\quad\mathbf x^{(1)}=\binom1{-i},
\qquad\lambda_2=-4-3i,\quad\mathbf x^{(2)}=\binom1i.}
$$
The distinct <eigenvalues> give two <linearly independent> solutions, and the general complex solution is
$$
\mathbf x(t)=a_1\mathbf x^{(1)}e^{\lambda_1t}+a_2\mathbf x^{(2)}e^{\lambda_2t}.
$$
Real solutions have $a_2=\overline{a_1}$. Equivalently,
$$
\mathbf x(t)=e^{-4t}\binom{C\cos3t-D\sin3t}{C\sin3t+D\cos3t},\qquad C,D\in\mathbb R.
$$
They spiral counterclockwise towards the origin, which is a <stable spiral>.
In a forced <linear differential equation>, <resonance in a differential equation> occurs when exponential forcing excites a mode with the same exponent as its <eigenvalue>, producing a polynomial factor such as $t$ multiplying that exponential. Merely matching an exponent is not enough: its projection onto that mode must be nonzero. Here $te^{\lambda_jt}$ still decays in absolute magnitude because the real part of $\lambda_j$ is negative; resonance refers to this extra factor, not necessarily to unbounded growth.
For each forcing vector, resolve it in the <eigenvector> basis:
$$
\binom{p_j}{q_j}=c_{1j}\binom1{-i}+c_{2j}\binom1i,
\qquad c_{1j}=\frac{p_j+iq_j}{2},\quad c_{2j}=\frac{p_j-iq_j}{2}.
$$
If $u_k$ is the coefficient of the $k$th <eigenvector>, its scalar equation is $\dot u_k=\lambda_k u_k+\sum_jc_{kj}e^{\lambda_jt}$. Multiplying by $e^{-\lambda_kt}$ and integrating shows that the $j=k$ contribution is $c_{kk}te^{\lambda_kt}$, whereas a $j\ne k$ contribution is $c_{kj}e^{\lambda_jt}/(\lambda_j-\lambda_k)$. Therefore the exact conditions for no resonant response are
$$
\boxed{p_1+iq_1=0,\qquad p_2-iq_2=0.}
$$
Under these conditions one <particular solution> is
$$
\mathbf x_{\rm p}(t)=\frac{c_{21}}{6i}\mathbf x^{(2)}e^{\lambda_1t}
-\frac{c_{12}}{6i}\mathbf x^{(1)}e^{\lambda_2t},
$$
which contains no factor of $t$. For a real forcing, $p_2=\overline{p_1}$ and $q_2=\overline{q_1}$, and the two boxed conditions are conjugate to each other.
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