How do you deal with the dark (negative) side of functions?

This was my inspiration: Negative Integer Number Base

We can use negative integers bases of -5 and below since we are told that $a, b, c. d. e<5$ and are all positive.

Hence we can take $f(-5)=1212$ :

$1212$ in base $\boxed{ -5}$ can be calculated as:

$1212 \div\boxed{ -5} = \color{#D61F06}{-242}$ , remainder $\color{#EC7300}{2}$

$\color{#D61F06}{-242} \div\boxed{ -5} =\color{#20A900}{ 49}$ , remainder $\color{#EC7300}{3}$

$\color{#20A900}{49} \div\boxed{ -5} = \color{#3D99F6}{-9}$ , remainder $\color{#EC7300}{4}$

$\color{#3D99F6}{-9} \div\boxed{ -5} = \color{#69047E}{2}$ , remainder $\color{#EC7300}{1}$

$\color{#69047E}{2} \div\boxed{ -5} = 0$ , remainder $\color{#EC7300}{2}$

Reading the remainders from bottom to top we get $1212 = 21432_{-5}$

$\color{#D61F06}{2} \cdot (-5)^4+\color{#3D99F6}{1} \cdot (-5)^3+\color{#20A900}{4} \cdot (-5)^2+\color{#EC7300}{3} \cdot (-5)^4+\color{#69047E}{2} \cdot (-5)^0 = 1212$

Therefore the polynomial is $\color{#D61F06}{2}x^4+\color{#3D99F6}{1}x^3+\color{#20A900}{4}x^2+\color{#EC7300}{3}x+\color{#69047E}{2}$

$\overline{abc} + \overline{cde} = \color{#D61F06}{2}\color{#3D99F6}{1}\color{#20A900}{4}+\color{#20A900}{4}\color{#EC7300}{3}\color{#69047E}{2} = \boxed{646}$

Lagrange interpolation can be used as well. We must have (the last three terms define the unique quadratic polynomial taking the desired values at $-7,-6,-5$ , and the first term makes $f(x)$ as general a quartic as possible): $f(x) \;= \; (\alpha x + \beta)(x+5)(x+6)(x+7) + \tfrac{4636}{2}(x+5)(x+6) - 2504(x+5)(x+7) + \tfrac{1212}{2}(x+6)(x+7)$ for some values of $\alpha$ and $\beta$ . The coefficients of $x^4$ , $x^3$ and $1$ in $f(x)$ are $\alpha$ , $18\alpha+\beta$ and $7352 + 210\beta$ respectively. Since each of these coefficients must be integers between $1$ and $4$ , inclusive, it follows that $\alpha$ and $\beta$ must both be integers, and indeed that $\alpha=2$ and $\beta=-35$ . Thus $f(x) \; = \; 2x^4 + x^3 + 4x^2 + 3x + 2$ and so the answer is $214 + 432 = 646$ .