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Find the marginal profit of a firm with a profit function:

$$P(q) = -192q + 88q^2 - 16q^3$$

I got

$$\frac{dP}{dq} = -192 + 176q - 48q^2$$

However, the solution reads

$$ \frac{dP}{dq}= -4(q-2)(q-4)(q-6)$$

Is this a typo, or am I missing something?

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The marginal profit you calculate is correct.

We can rearrange the solution of the problem you are given. This is equivalent to

$$ \frac{dP}{dq} = 192 -176q + 48q^2 -4q^3$$

This derivative has as primitive function the following profit function:

$$P(q) = c + 192q -88q^2 + 16q^3 - q^4$$

where $c$ is a constant (e.g. $c=0$).

This is clearly different from the original profit function in your problem. In consequence, either the profit function or the solution is wrong.

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