Answer
The solutions are $x = -1, -i, \text{ and } i$.
Work Step by Step
In this problem, we can see that certain terms can be grouped together. This means that we can factor this polynomial equation by grouping.
First, we group the terms:
$(x^3 + x^2) + (x + 1) = 0$
Next, we factor out the greatest common factor (GCF) of each group. For the first group, we can factor out an $x^2$:
$x^2(x + 1) + (x + 1) = 0$
The binomial $x + 1$ is a common factor so we factor it out:
$(x + 1)(x^2 + 1) = 0$
Use the Zero-Product Property by equating each factor to zero, then solve each equation.
First factor:
$x + 1 = 0$
$x = -1$
Second factor:
$x^2 + 1 = 0$
$x^2 = -1$
Take the square root of $-1$ to solve for $x$:
$x = \pm\sqrt {-1}$
The square root of $-1$ is $i$ so
$x = \pm i$
The solutions are $x = -1, \pm i$.
