Question 1162880
<br>
f(x) is a polynomial of degree 4 which is divisible by x^3.  So<br>
{{{f(x) = ax^4+bx^3}}}<br>
f(x)+1 is divisible by (x+1)^2 = x^2+2X+1.<br>
Perform the polynomial long division....<br><pre>

                              ax^2 + (b-2a)x + (3a-2b)
            ----------------------------------------------
   x^2+2x+1 ) ax^4 +    bx^3 +       0x^2   +   0x   +   1
              ax^4 + (2a)x^3 +       ax^2
              -------------------------------
                   (b-2a)x^3 -       ax^2 +      0x
                   (b-2a)x^3 + (2b-4a)x^2 + (b-2a)x
                   ---------------------------------------
                               (3a-2b)x^2 +  (2a-b)x +  1
                               (3a-2b)x^2 + (6a-4b)x + (3a-2b)
                               ---------------------------------
                                            (3b-4a)x + (1-3a+2b)</pre>
The remainder has to be zero, so<br>
{{{3b-4a = 0}}}
{{{1-3a+2b = 0}}}<br>
Solve the pair of equations (I leave that much to you) to find a=3 and b=4.<br>
So the function is<br>
{{{f(x) = 3x^4+4x^3}}}<br>
ANSWER:
{{{f(-4) = 3(-4)^4+4(-4)^3 = 3(256)+4(-64) = 768-256 = 512}}}<br>