1. The problem involves analyzing the function $$\mathbf{x}(t) = e^{-0.266t} \left\{ C_1 \begin{bmatrix} 0.732 \\ 1 \end{bmatrix} \cos 7.96t + C_2 \begin{bmatrix} 0.732 \\ 1 \end{bmatrix} \sin 7.96t \right\} + e^{-3.73t} \left\{ C_3 \begin{bmatrix} -2.73 \\ 1 \end{bmatrix} \cos 14.92t + C_4 \begin{bmatrix} -2.73 \\ 1 \end{bmatrix} \sin 14.92t \right\}$$ which represents a system with two exponentially decaying oscillatory components. 2. This type of function typically arises in solutions to systems of differential equations with complex eigenvalues, where each term is a product of an exponential decay and sinusoidal oscillations multiplied by constant vectors. 3. The exponential terms $e^{-0.266t}$ and $e^{-3.73t}$ indicate decay rates; the smaller decay rate $0.266$ corresponds to slower decay, and $3.73$ corresponds to faster decay. 4. The cosine and sine terms with frequencies $7.96$ and $14.92$ represent oscillations at those angular frequencies. 5. The vectors $\begin{bmatrix} 0.732 \\ 1 \end{bmatrix}$ and $\begin{bmatrix} -2.73 \\ 1 \end{bmatrix}$ scale the oscillations in two-dimensional space. 6. To graph or analyze this function, one would typically fix constants $C_1, C_2, C_3, C_4$ and plot the resulting vector function over time $t$. 7. The overall behavior is a combination of two damped oscillations with different decay rates and frequencies, resulting in a complex motion that decays to zero as $t \to \infty$. Final answer: The function describes a two-mode damped oscillatory system with decay rates $0.266$ and $3.73$, frequencies $7.96$ and $14.92$, and spatial vectors $\begin{bmatrix} 0.732 \\ 1 \end{bmatrix}$ and $\begin{bmatrix} -2.73 \\ 1 \end{bmatrix}$ respectively.