<human>: Give me the gist of the text in a nutshell

So I bought a used car from a used car dealership in PA. When we were test driving it, we drove it 3 times, and the dealer told myself and my father that it came with a 3 month, engine and transmission warranty. Obviously in a used car, that appealed to us greatly.

My father was out of town, so I put the car in my mother's name, but I commute to university so I had to get to class, and had to leave before the paperwork was signed and finalized. Well it seems the salesman neglected to tell my mother about the warranty, and she signed it away, thinking myself and my father agreed on no warranty. 

It is partly my fault because we let the salesman rush us. We wanted to wait for the end of the week, but he wanted to finalize the sale before the end of the month in August. The car also had to be inspected, which we were told took 2 days because the car was not a high end car. We bought the car on 8/31/2016.

When we got the car home, I started noticing things wrong. There is a coolant leak from a cheaply fixed radiator hose, and some steering component is worn out. I sent the car to transmission shop, and the transmission is shot and needs to be replaced for $2250. 

I know the warranty was signed away, but I know there is no way the dealer, which has it's own shop could sell this car and not know about the damage. The coolant was clearly cleaned up so I find it hard to believe they did not know about it. Do we have a leg to stand on, or did all hope go away with signing away the warranty?
<bot>: Bought a used car, told there would be a warranty when test driving, no warranty at sale 3 days later. Transmission needs replacing after 3 weeks. Is there anything we can do?