Hello from MrBillDoesMath!
Answer: 9
Discussion
By the Pythagorean theorem, the length of the third side (the hypotenuse) in the triangle with sides 4 and 6 is sqrt ( 4^2 + 6^2) = sqrt (52)
Applying Pythagoras to the left most triangle (sides, x and 6) gives:
x^2 + 6^2 = m^2 (I am calling the unknown side "m)
Finally, in the big right triangle (the one containing the two right triangles)
(x + 4) ^2 = (sqrt(52)) ^2 + m^2
OK! Subtract the first equation from the second:
(x+4)^2 = 52 + m^2
x^2 + 36 = m^2
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( x+4)^2 - x^2 - 36 = 52 + (m^2 - m^2) =>
x^2 + 8x + 16 - x^2 -36 = 52 =>
x^2 - x^2 + 8x + 16 -36 = 52 =>
0 + 8x - 20 = 52 =>
8x = 72 =>
x = 9
Thank you,
MrB