Suppose that a guitar company estimates that its monthly cost is $C(x)=400 x^{2}+600 x$ and its monthly revenue is $R(x)=-0.4 x^{3}+600 x^{2}-200 x+500$, where $x$ is in thousands of guitars sold. The profit is the difference between the revenue and the cost. What is the profit function, $P(x)$ ?

Step 1 ：Given the cost function $C(x) = 400x^2 + 600x$ and the revenue function $R(x) = -0.4x^3 + 600x^2 - 200x + 500$, we need to find the profit function $P(x) = R(x) - C(x)$.

Step 2 ：Subtracting the cost function from the revenue function, we get $P(x) = (-0.4x^3 + 600x^2 - 200x + 500) - (400x^2 + 600x)$.

Step 3 ：Simplifying the expression, we obtain the profit function $\boxed{P(x) = -0.4x^3 + 200x^2 - 800x + 500}$.