Constructing deterministic PDA for not regular language
Recipe in shorthand. Ignore $c$'s. Use a bottom fo stack symbol $Z$ that stays there. Keep the surplus of $a$'s on the stack. So $ZAAA$ (top of stack right) means three $a$'s more than $b$'s. If we read another $a$ push $A$, if we read another $b$ pop an $A$ (and move to another state to indicate we have now seen $b$).