A manufacturing model predicts profit: $P\left(t\right)=2t^2-11t-21$ dollars, where $t$ is in thousands of units. Factor $P\left(t\right)$ completely to identify production thresholds (in thousands of units) and interpret the roots.

• A. $P\left(t\right)=2(t-7)(t+\frac{3}{2})$, which gives the positive root $t=\frac{3}{2}$ indicating meaningful production threshold of $1,500$ units, the negative root is extraneous for production
• B. $P\left(t\right)=(2t-7)(t+3)$, which gives roots $t=3$ and $t=\frac{7}{2}$ and suggests different thresholds
• C. $P\left(t\right)=(t+7)(2t-3)$, producing roots $t=-7$ and $t=\frac{3}{2}$, implying the profit breaks even at $1,500$ units
• D. $P\left(t\right)=(t-7)(2t+3)$, which gives the positive root $t=7$ indicating meaningful production threshold of $7,000$ units, the negative root is extraneous for production

Answer: (D)