Explanation:

Let's analyze the given mappings to find the underlying pattern:\nFor $x=2$, the mapped value is $10$. We observe that $2^2 + 6 = 4 + 6 = 10$.\nFor $x=4$, the mapped value is $26$. We observe that $4^2 + 10 = 16 + 10 = 26$.\nFor $x=6$, the mapped value is $50$. We observe that $6^2 + 14 = 36 + 14 = 50$. \nThe pattern appears to be $x^2 + C$, where $C$ is a constant that depends on $x$. Let's look at the sequence of constants: $6, 10, 14$. This is an arithmetic progression with a common difference of $4$ ($10-6=4$, $14-10=4$).\nWe can express $C$ as a linear function of $x$. Notice that $C = 2x + 2$ fits the pattern:\nFor $x=2$, $C = 2(2) + 2 = 4 + 2 = 6$.\nFor $x=4$, $C = 2(4) + 2 = 8 + 2 = 10$.\nFor $x=6$, $C = 2(6) + 2 = 12 + 2 = 14$. \nSo, the general mapping rule is $f(x) = x^2 + (2x+2)$.\nNow, we need to find the number that $16$ is mapped to. Using the established rule for $x=16$: $f(16) = 16^2 + (2 \times 16 + 2)$ $f(16) = 256 + (32 + 2)$ $f(16) = 256 + 34$ $f(16) = 290$. \nLet's evaluate the options:\nOption (1) $132$: This does not follow the pattern $16^2 + (2 \times 16 + 2) = 290$.\nOption (2) $290$: This matches our calculated value.\nOption (3) $144$: This does not follow the pattern.\nOption (4) $268$: This does not follow the pattern. \nTherefore, $16$ is mapped to $290$. The correct option is (2) 290.