Ta có
$p(x) = x^4 + ax^3 + bx^2 + cx + d$
Khi đó
$p(1) = 1 + a + b + c + d$
$p(2) = 16 + 8a + 4b + 2c + d$
$p(3) = 81 + 27a + 9b + 3c + d$
Lại có
$p(9) = 9^4 + 729a + 81b + 9c + d$
$p(-5) = 5^4 -125a + 25b - 5c + d$
Do đó
$M = \dfrac{p(9) + p(-5)}{4}$
$= \dfrac{9^4 + 5^4 + (604a + 106b + 4c + 2d)}{4}$
Ta lại có
$604a + 106b + 4c + 2d = (49 + 49a + 49b + 49c + 49d) - (1536 + 768a + 384b +192c + 96d) + (3969 + 1323a + 441b + 147c + 49d) -2482$
$= 49(1 + a + b + c + d) - 96( 16 + 8a + 4b + 2c + d) + 49(81 + 27a + 9b + 3c + d) -2482$
$= 49p(1) - 96p(2) + 49p(3) - 2482$
Vậy
$M = \dfrac{9^4 + 5^4 + 49p(1) - 96p(2) + 49p(3) -2482}{4}$
$= \dfrac{9^4 + 5^4 + 49.827 - 96.1654 + 49.2481 - 2482}{4}$
$=2003$
Vậy $M = 2003$.
