Evaluate the first partial derivatives of the function at the given point. \[ \begin{array}{l} f(x, y)=4 x^{2} y+4 x y^{2} ; \quad(4,3) \\ f_{x}(4,3)=\square \\ f_{y}(4,3)=\square \end{array} \]

Step 1 ：Given the function \(f(x, y) = 4x^2y + 4xy^2\) and the point (4,3).

Step 2 ：The first partial derivative of a function with respect to x, denoted as \(f_x\), is the derivative of the function with respect to x, treating all other variables as constants.

Step 3 ：Similarly, the first partial derivative of a function with respect to y, denoted as \(f_y\), is the derivative of the function with respect to y, treating all other variables as constants.

Step 4 ：To find the first partial derivatives of the function \(f(x, y) = 4x^2y + 4xy^2\), we will first differentiate the function with respect to x to find \(f_x\), and then differentiate the function with respect to y to find \(f_y\).

Step 5 ：After finding the partial derivatives, we will evaluate them at the point (4,3).

Step 6 ：The first partial derivative of the function with respect to x is \(f_x = 8xy + 4y^2\).

Step 7 ：Evaluating \(f_x\) at the point (4,3) gives \(f_x(4,3) = 132\).

Step 8 ：The first partial derivative of the function with respect to y is \(f_y = 4x^2 + 8xy\).

Step 9 ：Evaluating \(f_y\) at the point (4,3) gives \(f_y(4,3) = 160\).

Step 10 ：Final Answer: The first partial derivative of the function \(f(x, y) = 4x^2y + 4xy^2\) with respect to x at the point (4,3) is \(\boxed{132}\), and with respect to y at the point (4,3) is \(\boxed{160}\).