Looking for low budget Motherboard Please Help!

I am a little confused here but I suspect that you may be chasing an upgrade that you don't need. 2x4GB of RAM is enough for most games but you don't have a discrete GPU so you likely aren't gaming. For most other applications and uses 2x4GB exceeds the maximum requirement. So why do you want to add more RAM?

If you are using the integrated graphics on the i5-4440 then HD 4600 is your iGPU.
https://ark.intel.com/products/75038/Intel-Core-i5-4440-Processor-6M-Cache-up-to-3_30-GHz

A GT 1030 will blow this out of the water.
http://gpu.userbenchmark.com/Compare/Intel-HD-4600-Desktop-125-GHz-vs-Nvidia-GT-1030/2168vsm283726

Now you talk about RAM but you don't provide full specs. Be aware that when you mix RAM the system will slow everything down to the speed of the slowest stick. Are you sure that the RAM is matched?

You don't state which country you will be purchasing in but I will assume U.S. But for $50 you find nothing new.
https://pcpartpicker.com/products/motherboard/#sort=price&page=1

You will have to look for a used mobo at this price, you can use this website to check for features and compatibility. Just click on the link to view/edit the build.

But maybe the better thing to do is to state what you want to achieve and then I can assist you in determining the best path to get there.


PCPartPicker part list: https://pcpartpicker.com/list/ptBmP3
Price breakdown by merchant: https://pcpartpicker.com/list/ptBmP3/by_merchant/

CPU: Intel - Core i5-4440 3.1GHz Quad-Core Processor (Purchased For $0.00)
Motherboard: Asus - B85M-E/CSM Micro ATX LGA1150 Motherboard ($85.19 @ Newegg Marketplace)
Memory: G.Skill - Ripjaws Series 16GB (4 x 4GB) DDR3-1600 Memory (Purchased For $0.00)
Total: $85.19
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2018-02-06 09:14 EST-0500

Okay, let's start with the RAM. Found some references to this PC and it might or might not take DDR3L (1.35V) RAM instead of the regular DDR3 (1.50V). So can you either supply the part number on the RAM or run this free program to determine what you have installed. I would not be mixing DDR3L and DDR3.
https://www.cpuid.com/softwares/cpu-z.html

You find the RAM voltage at the bottom of the "SPD" tab, this one is 1.50V DDR3
https://imgur.com/MIbyAYX

Next, I would not be using the 10600U (1333 MHz) ram as it will just slow your computer down regardless if you install 16GB. Very few applications need more than 2x4GB, I don't see a reason to spend money on another stick so you can go to 4x4GB. We can go down that path if you choose but just installing more RAM won't necessarily make things better.

So this leaves us with two paths, replace the mobo or keep using it. You will not be able to find a new mobo, case, and GPU for $100. A new GT 1030 which is a bottom end GPU will cost you almost this much.
https://pcpartpicker.com/products/video-card/#c=396&sort=price&page=1

You will be more at the $200 price range and these parts are as about as cheap as it gets.

PCPartPicker part list: https://pcpartpicker.com/list/dhcqq4
Price breakdown by merchant: https://pcpartpicker.com/list/dhcqq4/by_merchant/

CPU: Intel - Core i5-4440 3.1GHz Quad-Core Processor (Purchased For $0.00)
Motherboard: Asus - B85M-E/CSM Micro ATX LGA1150 Motherboard ($90.46 @ Newegg Marketplace)
Memory: G.Skill - Ripjaws Series 16GB (4 x 4GB) DDR3-1600 Memory (Purchased For $0.00)
Video Card: Gigabyte - GeForce GT 1030 2GB Low Profile Video Card ($81.99 @ SuperBiiz)
Case: Azza - SIRIUS ATX Mid Tower Case ($24.84 @ Amazon)
Total: $197.29
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2018-02-07 09:53 EST-0500

So if you want to continue down this path then you will have to get used components. To do this, scan the ads and plug the info into PCPartPicker to make sure it is compatible. For example, here are three Ebay components

https://www.ebay.com/itm/Gigabyte-Technology-GA-Z97-D3H-Intel-Motherboard/183059075706?epid=216102302&hash=item2a9f2bd27a:g:rFcAAOSwXkNaec6v#shpCntId
https://www.ebay.com/itm/4GB-KINGSTON-KVR16N11S8-4-DDR3-1600Mhz-12800U-Memory-DIMM/152890741752?epid=144642658&hash=item2398ff93f8:g:GeoAAOSwndZac74G
https://www.ebay.com/itm/New-in-Box-XFX-Radeon-Rx-550-4GB-PCI-E-Video-Card-RX-550P4SFG5/292439967221?epid=27013929510&hash=item4416c7c5f5:g:M54AAOSwlv9abhAJ

Here is the PCPartPicker build

PCPartPicker part list: https://pcpartpicker.com/list/DHykcY
Price breakdown by merchant: https://pcpartpicker.com/list/DHykcY/by_merchant/

CPU: Intel - Core i5-4440 3.1GHz Quad-Core Processor (Purchased For $0.00)
Motherboard: Gigabyte - GA-Z97-D3H ATX LGA1150 Motherboard ($40.00)
Memory: Kingston - 4GB (1 x 4GB) DDR3-1600 Memory ($20.00)
Memory: Kingston - 4GB (1 x 4GB) DDR3-1600 Memory (Purchased For $0.00)
Memory: Kingston - 4GB (1 x 4GB) DDR3-1600 Memory (Purchased For $0.00)
Memory: Kingston - 4GB (1 x 4GB) DDR3-1600 Memory (Purchased For $0.00)
Video Card: XFX - Radeon RX 550 4GB Video Card ($60.00)
Case: Azza - SIRIUS ATX Mid Tower Case ($24.84 @ Amazon)
Total: $144.84
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2018-02-07 10:12 EST-0500

Still breaks the budget.

Now for the other path of keeping the mobo. I think here you need to go to this webpage and determine what your system was shipped with. Just type in your SN and it will detail what it was shipped with.
https://www.acer.com/ac/en/US/content/support-product/4958;-;Aspire%20TC-605

The reason for this is that there is little spec information on this particular version and I am worried that you don't have room for a dual slot, full height GPU. As well, I am worried about the size of your PSU, one model ships with a 200W PSU, way too small. Also, need to know what PCI-e connectors the PSU has in case we need them.

Here is what I would do. Just because you own the RAM is not a reason to spend money for little return. If possible, keep your mobo, install 2x4GB of the 12800U (1600 MHz) RAM, purchase that RX 550 for $60 and you will have a decent little gamer. But of course, the choice is yours. Let me know how you want to proceed and I will try to help you get there.

Next, I would not be using the 10600U (1333 MHz) ram as it will just slow your computer down regardless if you install 16GB. Very few applications need more than 2x4GB, I don't see a reason to spend money on another stick so you can go to 4x4GB. We can go down that path if you choose but just installing more RAM won't necessarily make things better.

What OS are you running? 32-bit OS's will only use 4GB regardless of much is installed.