What is the maximum value of $4(x + 7)(2 - x)$, over all real numbers $x$?
I notice that this expression is a product of three factors: $4$, $(x + 7)$, and $(2 - x)$. To find the maximum value, I need to consider how each factor varies as $x$ changes. The factor $4$ is constant, so it does not affect the maximum value. The factor $(x + 7)$ is a linear function of $x$, which increases as $x$ increases. The factor $(2 - x)$ is also a linear function of $x$, but it decreases as $x$ increases. Therefore, the product of these two factors will be largest when they are both as large as possible, or as close to each other as possible. To find when that happens, I can set them equal to each other and solve for $x$: $(x + 7) = (2 - x)$. Simplifying, I get $2x = -5$, so $x = -\frac{5}{2}$. This means that the maximum value of the expression occurs when $x = -\frac{5}{2}$. To find the actual maximum value, I can plug this value of $x$ into the expression: $4(-\frac{5}{2} + 7)(2 + \frac{5}{2})$. Simplifying, I get $4(\frac{9}{2})(\frac{9}{2})$, which is $81$. 