Answer
The moment about the $x$-axis is
$$
\begin{aligned}
M_{x}&=\sum_{i=1}^{4} m_{i} y_{i} \\
&=24
\end{aligned}
$$
The moment about the $y$-axis is
$$
\begin{aligned}
M_{y}&=\sum_{i=1}^{4} m_{i} x_{i} \\
&=-5
\end{aligned}
$$
The center of mass is
$$
\begin{aligned}
(\overline{x}, \overline{y})& =\left(\frac{M_{y}}{m}, \frac{M_{x}}{m}\right) \\
&=\left(\frac{M_{y}}{\sum_{i=1}^{4} m_{i}}, \frac{M_{x}}{\sum_{i=1}^{4} m_{i}}\right)\\
&=\left(\frac{-5}{18}, \frac{24}{18}\right) \\
&=\left(\frac{-5}{18}, \frac{4}{3}\right) .
\end{aligned}
$$
Work Step by Step
the system:
$$
\begin{array}{l}{m_{1}=5, m_{2}=4, m_{3}=3, m_{4}=6} \\ {P_{1}(-4,2), P_{2}(0,5), P_{3}(3,2), P_{4}(1,-2)}\end{array}
$$
The moment about the $x$-axis is
$$
\begin{aligned}
M_{x}&=\sum_{i=1}^{4} m_{i} y_{i} \\
&=m_{1} y_{1}+m_{2} y_{2}+m_{3} y_{3}+m_{4} y_{4} \\
&=5\cdot (2)+4\cdot (5)+3\cdot (2) +6\cdot (-2) \\
&=24
\end{aligned}
$$
The moment about the $y$-axis is
$$
\begin{aligned}
M_{y}&=\sum_{i=1}^{4} m_{i} x_{i} \\
&=m_{1} x_{1}+m_{2} x_{2}+m_{3} x_{3} +m_{4} x_{4}\\
&=5\cdot (-4)+4\cdot (0)+3\cdot (3)+6\cdot (1) \\
&=-5
\end{aligned}
$$
The center of mass is
$$
\begin{aligned}
(\overline{x}, \overline{y})& =\left(\frac{M_{y}}{m}, \frac{M_{x}}{m}\right) \\
&=\left(\frac{M_{y}}{\sum_{i=1}^{4} m_{i}}, \frac{M_{x}}{\sum_{i=1}^{4} m_{i}}\right)\\
&=\left(\frac{M_{y}}{m_{1} +m_{2} +m_{3}+m_{4}}, \frac{M_{x}}{m_{1} +m_{2} +m_{3}+m_{4} }\right)\\
&=\left(\frac{M_{y}}{5+4 +3+6}, \frac{M_{x}}{5+4 +3+6}\right)\\
&=\left(\frac{-5}{18}, \frac{24}{18}\right) \\
&=\left(\frac{-5}{18}, \frac{4}{3}\right) .
\end{aligned}
$$
