Solution:

Given that $A=\left[\begin{array}{lll}3 & 2 & 5 \\ 4 & 1 & 3 \\ 0 & 6 & 7\end{array}\right]$, to express as sum of symmetric matrix $P$ and skew symmetric matrix $Q$.
$A=p+Q$
Where $P=\frac{1}{2}\left(A+A^{\prime}\right)$ and $Q=\frac{1}{2}\left(A-A^{\prime}\right)$
First we need to find $A^{2}$
$A^{\prime}=\left[\begin{array}{lll}
3 & 4 & 0 \\
2 & 1 & 6 \\
5 & 3 & 7
\end{array}\right]$
Using above mentioned formulas
$\begin{array}{l}
P=\frac{1}{2}\left(A+A^{\prime}\right) \\
\Rightarrow \frac{1}{2}\left(\left[\begin{array}{lll}
3 & 2 & 5 \\
4 & 1 & 3 \\
0 & 6 & 7
\end{array}\right]+\left[\begin{array}{lll}
3 & 4 & 0 \\
2 & 1 & 6 \\
5 & 3 & 7
\end{array}\right]\right) \\
\Rightarrow \frac{1}{2}\left[\begin{array}{ccc}
6 & 6 & 5 \\
6 & 2 & 9 \\
5 & 9 & 14
\end{array}\right]
\end{array}$
$\Rightarrow\left[\begin{array}{ccc}3 & 3 & \frac{5}{2} \\ 3 & 1 & \frac{9}{2} \\ \frac{5}{2} & \frac{9}{2} & 7\end{array}\right]$
$Q=\frac{1}{2}\left(A-A^{\prime}\right)$
$\Rightarrow \frac{1}{2}\left(\left[\begin{array}{lll}3 & 2 & 5 \\ 4 & 1 & 3 \\ 0 & 6 & 7\end{array}\right]-\left[\begin{array}{lll}3 & 4 & 0 \\ 2 & 1 & 6 \\ 5 & 3 & 7\end{array}\right]\right)$
$\Rightarrow \frac{1}{2}\left[\begin{array}{ccc}0 & -2 & 5 \\ 2 & 0 & -3 \\ -5 & 3 & 0\end{array}\right]$
$\Rightarrow\left[\begin{array}{ccc}0 & -1 & \frac{5}{2} \\ 1 & 0 & \frac{-3}{2} \\ \frac{-5}{2} & \frac{3}{2} & 0\end{array}\right]$
Now $A=P+Q$
$\Rightarrow\left[\begin{array}{ccc}3 & 3 & \frac{5}{2} \\ 3 & 1 & \frac{9}{2} \\ \frac{5}{2} & \frac{9}{2} & 7\end{array}\right]+\left[\begin{array}{ccc}0 & -1 & \frac{5}{2} \\ 1 & 0 & \frac{-3}{2} \\ \frac{-5}{2} & \frac{3}{2} & 0\end{array}\right]$
$\Rightarrow\left[\begin{array}{lll}3 & 2 & 5 \\ 4 & 1 & 3 \\ 0 & 6 & 7\end{array}\right]$