$2^2$

Let $|AB| = |FG| = a$ and $|BC| = |CD| = b$ . Then $||EF| = a - b$ and $|AC| = a + b$ . By Pythagoras, we then have that

$|FG|^{2} + |EF|^{2} = 34^{2} \Longrightarrow a^{2} + (a - b)^{2} = 1156$ and

$|EF|^{2} + |AC|^{2} = 46^{2} \Longrightarrow (a - b)^{2} + (a + b)^{2} = 2116 \Longrightarrow (a^{2} - 2ab + b^{2}) + (a^{2} + 2ab + b^{2}) = 2116 \Longrightarrow a^{2} + b^{2} = 1058$ .

Now $((a - b)^{2} + (a + b)^{2}) - (a^{2} + (a - b)^{2}) = 2116 - 1156 \Longrightarrow (a + b)^{2} - a^{2} = 960 \Longrightarrow 2ab + b^{2} = 960$ .

But by Pythagoras $x = |AD|^{2} = |AC|^{2} + |CD|^{2} = (a + b)^{2} + b^{2} = a^{2} + 2ab + b^{2} + b^{2} = (a^{2} + b^{2}) + (2ab + b^{2}) = 1058 + 960 = \boxed{2018}$ .