A cubic polynomial will fit.
Let f(x)=ax3+bx2+cx+d,
(1) f(-2)=2=-8a+4b-2c+d,
(2) f(-1)=-1=-a+b-c+d,
(3) f(1)=5=a+b+c+d,
(4) f(3)=67=27a+9b+3c+d,
(5) f(-1)+f(1)=4=2b+2d, b+d=2, d=2-b, a+c=3, c=3-a.
(6) -8a+4b-6+2a+2-b=2, -6a+3b=6, -2a+b=2, b=2+2a (from (1)),
(7) 27a+9b+9-3a+2-b=67, 24a+8b=56, 3a+b=7 (from (4)),
3a+2+2a=7, 5a=5, a=1, b=4, c=2, d=-2.
f(x)=x3+4x2+2x-2 is the polynomial.