|Your cheapest and easiest solution is to custom rig up a case fan (e.g. as little as $10 or less) inside the case so that it blows on the graphics card.|
"ATX cards face to the right side."
Same goes for BTX cards, as seen from the top of the card and the bracket end.
A Dell Dimension E520 with BTX form factor
Note the components are on the RIGHT side.
As jam has said, the mboard has upside down design.
Comparison microBTX vs microATX mboard
When an ATX or MATX mboard is installed, the card slots are at the bottom of a tower case - yours are at the top, whether it's a BTX form factor mboard or not.
The form factor dimensions of the mboards are on the right.
If yours is one of the BTX sizes, if you want a larger case, you need a BTX compatible case.
"The BTX form factor motherboards are incompatible with most of the ATX form factor cases and vice-versa. Some cases such as the Cooler Master Series (Stackers) support a varying range of motherboard types such as ATX, BTX, Mini-ATX and so forth. However, all connectors are compatible, including power supplies, PCI cards, processors, RAM, hard drives, etc."
Gateway Sx2800-03 support
Intel® Graphics Media Accelerator X4500 (Intel® GMA X4500)
PCI Express® 2.0 ×16 low-profile graphics card support
Power Supply 220 watt
The original graphics card has a low-profile (shorter height) bracket.
If you get a new case, you may need a graphics card that has a regular bracket, unless you get a small form factor case.
Your present power supply is probably a smaller than standard physical size. It may not fit in the new case.
Standard (PS/2) power supply size - 86mm high, 150mm wide, 140mm deep, or 3 3/8" h x 5 7/8" w x 5 1/2" d , or very close to that, though the depth can be more or less for some PSs.
Even if it does fit, if you were to get a small form factor case, if you upgrade the graphics card, you will probably need a power supply with more capacity.
Your power supply must have at least the minimum capacity required to support a system with the graphics card you are using installed, or the max graphics card you might install in the future.
(Onboard video - video built into the mboard - IS NOT A CARD ! )
If that info is not in the ad for the video card, you can go to the video card maker's web site and look up the specs for the model - often under system requirements - the minimum PS wattage, and, more important, the minimum amperage the PS must supply at 12v is stated. If you don't find that, any card with the same video chipset including any letters after the model number has very similar minimum PS requirements. Some power supplies have two or more +12v ratings - in that case, add those ratings to determine the total +12v current capacity.
If you're a gamer...
In most if not all cases, the max capacity rating of the PS is an intermittent rating. It's recommended that you do not load your PS to any more that 80% of that rating if you are going to be using something that puts a constant load on it, such as playing a recent game for hours on end. In that case, you multiply the min capacity stated for the system with the particular video chipset on the card by 1.25 to find the min. capacity of the PS you should get.
Don't buy an el-cheapo PS.
See response 3 in this: