I have a house built prior to 1960. Currently, the drainage line from the kitchen sink is backing up. We snaked over 40 ft. still didn't unclog stoppage. Water is coming out the washout on the side of the house, so no flooding in the house but the laminate floors seem to be separating and warping in different areas through the kitchen and dining area. Could this be from a broken drainage line underneath the slab? If so, what are the options to fix?

Have the entire area under the house video'ed to find out what the trouble really is.

40 ft ? Is that the entire distance under the house ? Or do you need to go 50 or 80 feet, if so a longer snake is needed.

Chuck gave you very good advice, However, perhaps it's not necessary to tear up the slab. I had the same problem in my 1955 home,
Let me explain a bit more about my drainage problem and how I repaired it. 54 year old house. The builder used sub standard cast iron in the drainage. The 4" house main was completely ate up on the bottom of the pipe , blocked with roots and welling up in my living room floor. I was sure that the main had broke in two and I would have to jack hammer up the floor and tunnel under the foundation to transition to PVC, pick up the back bath and move the main outside the house but I had to know what was going on under the slab so a friend suggested running a camera down there. There was no distinct break in the main, however,it was completely ate up on the bottom. That's when Rooter Man of Sarasota and Gulf Coast Florida suggested relining the main, (see image). It took a day to clear the roots and pressure jet clean the main. Next day they came back and relined the main with epoxy. I now have a sewer main that will out last me, (50 year guarantee) and the best part is that I didn't have to, tear up my house and the cost was under half of what I figured to take the main outside around the house. Another bonus was that it was done without tearing up or disturbing my house. When they were finished I kept a piece of the epoxy liner to check it out. It was white and had the same thickness of Schedule#40 PVC.
I was amazed at the equipment RooterMan had outside my place. The equipment he had cost upwards of $100,000.00. Hi tech stuff that took a technician to run it. There were 3 men and 2 trucks to do the job. And it was completed in two days. Lottsa difference from the old days when I went out on a sewer call with a Ridged K-60 Sewer Machine and a ladder to get to the roof vent. I just wanted to point out that there's another option besides replacing drainage that the years have ruined. If your pipes are a candidate for a reline job I strongly suggest you look into it before you decide to replace the drainage pipes. I did and saved a bundle in addition to keeping my home intact. Just thought I would share that with you all. Regards and thank you for rating my answer, Tom