Polynomial Problem

We have $a_0+a_1+\ldots + a_n = 4,$ implying $\max{a_0, a_1, \ldots, a_n} < 5.$

$P(5) = a_0 + 5 \cdot a_1 + 5^2 \cdot a_2 + \ldots + 5^n \cdot a_n$

thus the unique base $5$ representation of $152$ . Since $152 = 1102_5,$

$P(x)=x^3+x^2+2.$

$P(6)=216+36 + 2 = \boxed{254}.$

When $x=1$

$a_0+a_1+a_2................................................+a_n=4$

Either except four of these or less, $a_0+a_1+a_2.......................................................a_n$ all equal to $0$

$P(5)=152$

Hence the only possible value of constant term is 2.

maximum three terms are possible and all other are 0.

Let $5^x + 5^y=152-2$ where x and y are natural no.s

The only possible pair is 2,3.

Hence the $P(x)=x^3+x^2+2$

$P(6)=254$

The coefficients are non-negative: thus positive or 0. Use the method of undetermined coefficients.