College Algebra, Chapter 5, Review Exercise, Section Review Exercise, Problem 18

Determine the domain of the function $g(x) = \log (2 + x - x^2)$
If the given is a Logarithmic Function, then we want

$
\begin{equation}
\begin{aligned}
2 + x - x^2 &> 0 && \text{Model}\\
\\
x^2 - x - 2 &< 0 && \text{Multiply both sides by } -1\\
\\
(x+1)(x-2) &< 0 && \text{Factor}
\end{aligned}
\end{equation}
$

The factors on the left hand side are $x + 1$ and $x - 2$, these factors are 0 when $x$ is $-1$ and $2$ respectively. These numbers divide the number line
into intervals
$(-\infty,-1)(-1,2)(2,\infty)$
By testing some points in the interval,


Thus $(x + 1)(x-2) < 0$ at interval $(-1,2)$. Therefore, the domain of $g$ is $(-1,2)$