@mjt4283 Any card you can buy these days will run dual displays even in Extend mode, i.e. independently with different content on each display, and certainly will run in Mirror/Duplicate mode if you want to run them that way.  However, given that the U2410 has a resolution of 1920x1200 (16:10 aspect ratio) whereas most TVs have a resolution of 1920x1080 or some other 16:9 aspect ratio, if you choose to run them in Mirror/Duplicate mode, you'll end up with either stretching or black bars on one display or the other, because the displays simply aren't the same proportional shape.

In terms of that specific card, it has DVI and DisplayPort outputs, not an HDMI output.  There are DisplayPort to HDMI adapters, and since DisplayPort is also capable of carrying audio, you should still be able to send audio to an HDMI display/TV that way, but I don't have direct experience with that card.  But honestly given that DVI is obsolete and there are also DisplayPort to DVI adapters/cables, you might want to look at a video card that has a more modern and convenient port setup just for futureproofing, like one that has DisplayPort and HDMI outputs.  They might cost a bit more, though.  If you do go that route, you can use a DP to DVI adapter for your U2410 and a regular HDMI cable for the TV.

Lastly, it's impossible to give an answer as to whether a card will work when you're not even specifying what system you plan to install it into, although the fact that the card is half-height, single-width, and can be run purely on PCIe slot power rather than requiring auxiliary power admittedly makes its compatibility pretty high.

@mjt4283  That card looks like a better and simpler fit for your purposes.  Just be aware that with that card's port setup, you may be limited to 1920x1200 on your displays in case that poses a concern around future upgradeability for you.  Technically the DVI output might support dual-link DVI, which can run 2560x1600, but that's not clear from the specs, and you won't find any modern displays that support dual-link DVI inputs anyway.  The HDMI output may also go higher, but again it's not clear from the specs what revision of HDMI it supports, and if you'll be permanently allocating it to your TV, then that might be a moot point anyway.  If you think you might want to get a higher resolution display at some point, you might still want to look for GPUs that have DisplayPort and HDMI outputs instead, but otherwise that card looks fine for your current purposes.  A 40-foot HDMI run is quite long though, even for 1080p, and long runs like that are one of the few cases where it can actually be sensible rather than a waste of money to spend a bit more for a quality cable in order to get a reliable signal.  You might also want to look into "active" HDMI cables, which have built-in circuitry to amplify the signal, like Redmere HDMI cables.  They cost more, but the active circuitry means they can reliably achieve longer runs while remaining much slimmer than a regular passive cable would need to be in order to maintain signal integrity over that distance.  Also be aware that active HDMI cables are often directional, meaning that it matters which end you connect to the source vs. the TV, so be careful about that, otherwise the cable won't work at all.