The profit function $\mathrm{P}(\mathrm{x})$, in dollars, for a product is given by the equation
\[
P(x)=-0.4 x^{3}+132 x^{2}-3200 x-120,000
\]
where $\mathrm{x}$ is the number of units produced and sold. The break-even occurs when 50 units are produced and sold.
(a) Use synthetic division to find a quadratic factor of $\mathrm{P}(\mathrm{x})$.
(b) Use factoring to find a number of units other than 50 that gives break-even for the product, and verify your answer graphically.
(a) A quadratic factor of $P(x)$ is $-0.4 x^{2}+112 x+2400$.
(b) The number of units other than 50 that gives break-even for the product is

Step 1 ：Given the profit function \(P(x) = -0.4x^3 + 132x^2 - 3200x - 120000\), where \(x\) is the number of units produced and sold. The break-even occurs when 50 units are produced and sold.

Step 2 ：Use synthetic division to find a quadratic factor of \(P(x)\). The coefficients of \(P(x)\) are -0.4, 132, -3200, and -120000. The root is 50. After synthetic division, the new coefficients are -0.4, 112, 2400, and 0.

Step 3 ：The quadratic factor of \(P(x)\) is \(-0.4x^2 + 112x + 2400\).

Step 4 ：To find the number of units other than 50 that gives break-even for the product, we need to find the roots of this quadratic equation. The roots of the quadratic equation are the values of \(x\) for which \(P(x) = 0\).

Step 5 ：The roots of the quadratic factor are -20 and 300. However, the number of units produced and sold cannot be negative, so the only valid root is 300.

Step 6 ：Final Answer: The number of units other than 50 that gives break-even for the product is \(\boxed{300}\).