If the school suffered a loss and the world is not improving then teachers lose their jobs.
a = "The school suffered a loss"
b = "The world is improving"
c = "Teachers lose their jobs"
I got the answer of $(a \wedge\neg b \wedge\neg c)$ but that doesn't sound right when putting it back into English.
On
The linguistic carryover only goes so far. Your assertion is logically correct: $$ \begin{align*} \neg[(a\land\neg b)\to c] &\equiv \neg[\neg(a\land\neg b)\lor c]\\[0.5em] &\equiv (a\land\neg b)\land\neg c\\[0.5em] &\equiv a\land\neg b\land\neg c. \end{align*} $$ Now, how exactly you choose to phrase this is up to you a fair amount. I might express this as, "The school suffered a loss, the world is not improving, and teachers did not lose their jobs." Say it how you want, but just know you are logically correct. The linguistic mechanics are a more subtle challenge, and you can really only do but so much.
On
Daniel's answer covers why (formally) we get $a\wedge\neg b\wedge\neg c$ as the negation of $(a\wedge\neg b)\to c$.
However, the counterintuitive thing about "implies" statements is that when you negate one you don't get another "implies" statement, you get a description of the set of possible counterexamples.
This sounds a bit strange when translated back into English, and partly the reason is that $a\to b$ mathematically is talking about an $a$ and a $b$ with definite truth values, where as when you say "if X then Y" in English, there is the implication that the status of X and Y are unknown, but in all situations where X is true, Y is also true. So there is an implicit "for all" in the English statement you wanted to negate - and if you negate "for all...", you get "for some..." or "there exists...".
The point of all this is that if you want the negation of an "implies" statement to sound natural in English, you can add a "sometimes". So a natural negation of
If the school suffered a loss and the world is not improving then teachers lose their jobs.
would be
Sometimes the school suffers a loss and the world is not improving and teachers do not lose their jobs.
Your original statement is: $\lnot a\lor b\lor c$, or in English: "The school does not suffer loss, or the world is improving, or the teachers lose their jobs".
The negation is, as you say: $a\land\lnot b\land\lnot c$, i.e. "School suffers loss, and the world is not improving, but the teachers don't lose their jobs".