Functions Again?

Two methods to solve this:

Method 1 (The lazy but quick method)

Notice that for $x=-1,2,-3,4$

$f(x) = -x^2$

This implies that $f(x) +x^2 = 0$ has roots $x=-1,2,-3,4$

$\implies f(x) + x^2 = (x+1)(x-2)(x+3)(x-4)\\ f(x) = (x+1)(x-2)(x+3)(x-4) - x^2$

Check all the available options:

$f(1) = 23\\ f(0) = 24\\ f(-2) = -28\\ f(0) + f(-2) = 24 + (-28) = -4$

Therefore, it should be $\boxed{\text{All options are correct}}$ (I'm assuming that there is an error in the options)

---

Method 2 (The more tedious but easier to understand method)

Let $f(x) = x^4 +ax^3 + bx^2+cx+d$ (Monic means that the leading coefficient is 1)

Substitute the values of $x$ and $f(x)$ into the equation above, and you will get:

$-a+b-c+d=-2\\ 8a+4b+2c+d=-20\\ -27a+9b-3c+d = -90\\ 64a+16b+4c+d = -272$

Solve this system of linear equations, and you should get $a=-2,b=-14,c=14,d=24$

Therefore, $f(x) = x^4-2x^3-14x^2+14x+24$

Check all the available options:

$f(1) = 23\\ f(0) = 24\\ f(-2) = -28\\ f(0) + f(-2) = 24 + (-28) = -4$

Therefore, it should be $\boxed{\text{All options are correct}}$ (I'm assuming that there is an error in the options)