FLEX is an Airbus term for reduced thrust takeoff (flexible temperature).
During the takeoff phase FADEC calculates and commands a fixed takeoff thrust setting based upon an assumed temperature.
Quoting the Jeppesen A320 Airport Analysis manual
<font face="Verdana, Arial, Helvetica" size="2">Reduced Thrust Takeoff (Flexible Temperature)
Use of reduced thrust levels is optional but will ensure operating efficiency, reliability and engine service life. This "Flex" takeoff thrust is based upon an "Assumed temperature" (the maximum ambient temperature that will provide certificated takeoff performance for the given airport/runway and aircraft gross weight).
Flex Takeoff is not permitted when the runway is contaminated by standing water, slush, ice, or snow. Additionally, Flex takeoff is not recommended with tailwinds.
[list=1][*]Enter runway analysis with actual weight and wind. If actual weight is not shown, and interpolation is not desired, use the next highest weight shown in the analysis.[*]Read the maximum temperature shown in the box, or else interpolate[*]Apply temperature corrections shown in section iii as appropriate to determine the Corrected Temperature (CT). CT = maximum temperature + correction.[*]If CT is higher than OAT and TREF, use CT as flexible temperature, limit CT to ISA+45ºC.[*]If CT is below either OAT or TREF, the flexible takeoff is not allowed, and full takeoff thrust must be performed.[*]Takeoff speeds should be based on the flexible temperature (CT), and must exceed the minimum speeds computed for the actual OAT. If the minimum speeds exceed those shown at the flexible temperature, then a full thrust takeoff should be performed.[/list=a]</font>
The approach of how tailwind is used varies between operators, some allow FLEX takeoff with, some don’t, MEL items should be taken into account, the local weather (no wind shear), and crew judgment.
TREF is the engine flat rated temperature, A320 ISA+45ºC, A330 ISA+43ºC
Let me know if you would like to know what the corrections are for QNH, Anti-ice, and Bleeds are.