The profit, `P`, in dollars, from manufacturing and selling `x` custom phone cases per day is modeled by the equation `…

To find the number of cases for which the manufacturer breaks even, the profit `P` must be set to 0. This gives the equation `0 = -x² + 50x - 400`. To solve for `x`, we can first multiply the entire equation by -1 to make the `x²` term positive, which yields `x² - 50x + 400 = 0`. This is a quadratic equation that can be factored. We need to find two numbers that multiply to 400 and add up to -50. These two numbers are -10 and -40.

Therefore, the factored form of the equation is `(x - 10)(x - 40) = 0`. The solutions for `x` are 10 and 40. The question asks for the smaller number of cases, which is 10.

Choice A is correct.

Choice B is incorrect because it is the larger of the two break-even points.

Choice C is incorrect; it is the number of cases that would maximize profit, found by calculating the x-coordinate of the parabola's vertex (`-b/(2a)`), not a break-even point.

Choice D is incorrect; it is the coefficient of the x-term in the model, not a solution to the equation `P=0`.