解:?$(2)$?∵?$f(0)=-2$?,∴?$c=-2$?
∵?$f(1)=5$?,∴?$a+b+4-2=5$?,∴?$a+b=3$?
∴?$f(-1)=-a-b-4-2=-(a+b)-6=-9$?
?$(3)$?∵?$f(5)+f(-5)=6$?
∴?$5^5a+5^3b+20+c-5^5a-5^3b-20+c=6$?,∴?$c=3$?
∵?$f(2)=8$?,∴?$2^5a+2^3b+8+3=8$?,∴?$2^5a+2^3b=-3$?
∴?$f(-2)=-2^5a-2^3b-8+3 =-(2^5a+2^3b)-8+3 = -(-3)-8+3 =-2$?