Is it really correct?

Let the additional root of $f(x)$ be $\large \lambda$ .

Using Vieta's Formula, product of roots of $g(x)$ is $\large -c$ and product of roots of $f(x)$ is $\large c$ .

$\therefore \quad \large -c \times \lambda = c \implies \lambda = -1$

Also, Using Vieta's Formula, sum of roots of $g(x)$ is $\large -a$ and sum of roots of $f(x)$ is $\large -1$ .

$\therefore \quad \large -a + \lambda = -1 \implies a = 0$

hence $\large f(x) = (x+1)g(x)$

$\large \therefore f(1) = 2g(1)$

$\implies \large 102 + b + c = 2 \times (1 + b + c)$

$\large \implies b + c = 100$

$\large f(1) = 2 \times (1 + 100) = \boxed{202}$