To solve the given matrix expression, we need to perform matrix multiplication and then square the resulting matrix. The given matrices are:

\[ A = \begin{pmatrix} 3 & 3 \\ 5 & 4 \end{pmatrix} \]
\[ B = \begin{pmatrix} 2 & 1 \\ 6 & 5 \end{pmatrix} \]

First, we multiply matrices \( A \) and \( B \):

\[ A \cdot B = \begin{pmatrix} 3 & 3 \\ 5 & 4 \end{pmatrix} \cdot \begin{pmatrix} 2 & 1 \\ 6 & 5 \end{pmatrix} \]

To find the element in the first row and first column of the product matrix:
\[ (3 \cdot 2) + (3 \cdot 6) = 6 + 18 = 24 \]

To find the element in the first row and second column of the product matrix:
\[ (3 \cdot 1) + (3 \cdot 5) = 3 + 15 = 18 \]

To find the element in the second row and first column of the product matrix:
\[ (5 \cdot 2) + (4 \cdot 6) = 10 + 24 = 34 \]

To find the element in the second row and second column of the product matrix:
\[ (5 \cdot 1) + (4 \cdot 5) = 5 + 20 = 25 \]

So, the product matrix \( AB \) is:
\[ AB = \begin{pmatrix} 24 & 18 \\ 34 & 25 \end{pmatrix} \]

Next, we need to square this resulting matrix:

\[ (AB)^2 = \begin{pmatrix} 24 & 18 \\ 34 & 25 \end{pmatrix} \cdot \begin{pmatrix} 24 & 18 \\ 34 & 25 \end{pmatrix} \]

To find the element in the first row and first column of the squared matrix:
\[ (24 \cdot 24) + (18 \cdot 34) = 576 + 612 = 1188 \]

To find the element in the first row and second column of the squared matrix:
\[ (24 \cdot 18) + (18 \cdot 25) = 432 + 450 = 882 \]

To find the element in the second row and first column of the squared matrix:
\[ (34 \cdot 24) + (25 \cdot 34) = 816 + 850 = 1666 \]

To find the element in the second row and second column of the squared matrix:
\[ (34 \cdot 18) + (25 \cdot 25) = 612 + 625 = 1237 \]

So, the squared matrix \((AB)^2\) is:
\[ (AB)^2 = \begin{pmatrix} 1188 & 882 \\ 1666 & 1237 \end{pmatrix} \]

Thus, the final result is:
\[ \begin{pmatrix} 1188 & 882 \\ 1666 & 1237 \end{pmatrix} \]