Problem (Algebra):
The functions $f$ and $g$ satisfy

$$
\begin{aligned}
& f(x)+g(x)=3 x+5 \\
& f(x)-g(x)=5 x+7
\end{aligned}
$$

for all values of $x$. Determine the value of $2 f(2) g(2)$.

Solution:
First, we add the two given equations to obtain

$$
(f(x)+g(x))+(f(x)-g(x))=(3 x+5)+(5 x+7)
$$

or $2 f(x)=8 x+12$ which gives $f(x)=4 x+6$.

Since $f(x)+g(x)=3 x+5$, then $g(x)=3 x+5-f(x)=3 x+5-(4 x+6)=-x-1$.

(We could also find $g(x)$ by subtracting the two given equations or by using the second of the given equations.)

Since $f(x)=4 x+6$, then $f(2)=14$.

Since $g(x)=-x-1$, then $g(2)=-3$.

Therefore, $2 f(2) g(2)=2 \times 14 \times(-3)=-84$.

Since the two given equations are true for all values of $x$, then we can substitute $x=2$ to obtain

$$
\begin{aligned}
& f(2)+g(2)=11 \\
& f(2)-g(2)=17
\end{aligned}
$$

Next, we add these two equations to obtain $2 f(2)=28$ or $f(2)=14$.

Since $f(2)+g(2)=11$, then $g(2)=11-f(2)=11-14=-3$.

(We could also find $g(2)$ by subtracting the two equations above or by using the second of these equations.)

Therefore, $2 f(2) g(2)=2 \times 14 \times(-3)=-84$.

Final answer: -84