An algebra problem by atishay jain

Since $a$ , $b$ , $c$ , and $d$ are the roots of $f(x)$ , then

$\begin{aligned} f(x) & = x^4 - (a+b+c+d)x^3 + (ab+ac+ad+bc+bd+cd)x^2 - (abc + abd + acd + bcd)x + abcd \\ \implies f(-1) & = 1+ {\color{#3D99F6}a} +b+c+d +{\color{#3D99F6}a}b+{\color{#3D99F6}a}c+{\color{#3D99F6}a}d+bc+bd+cd + {\color{#3D99F6}a}bc + {\color{#3D99F6}a}bd + acd + bcd + {\color{#3D99F6}a}bcd \\ & = 1 + {\color{#3D99F6}a}\big[(1+{\color{#D61F06}b} +c+d+{\color{#D61F06}b}c +{\color{#D61F06}b}d + {\color{#D61F06}b}cd)\big] + {\color{#D61F06}b} +{\color{#D61F06}b}c +{\color{#D61F06}b}d + {\color{#D61F06}b}cd + c + cd + d + acd \\ & = 1 + a\big[(1+c+d+{\color{#D61F06}b}(1+ c +d + cd)\big] + {\color{#D61F06}b}(1 +c +d + cd) + c(1 + d) + d(1 + ac) \\ & = 1 + a\big[(1+c+d+b(1+ c)(1+d)\big] + b(1+c)(1+d) + c(1 + d) + d(1 + ac) \\ & = 1 + 98 \\ & = \boxed{99} \end{aligned}$