Answer
The solutions are $ -2, -i\sqrt {5}, \text{ and } -i\sqrt {5}$.
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:
$(x^3 + 2x^2) + (5x + 10) = 0$
Next, we factor out the greatest common factor (GCF) of each group. For the first group, we can factor out an $x^2$:
$x^2(x + 2) + (5x + 10) = 0$
Now, we can factor out a $5$ in the second group:
$x^2(x + 2) + 5(x + 2) = 0$
The binomial $x + 2$ is a common factor of the two groups, factor it out to obtain:
$(x^2 + 5)(x + 2) = 0$
Solve the equation by equating each factor to $0$, then solve each equation.
First factor:
$x^2 + 5 = 0$
$x^2 = -5$
$x = \pm \sqrt{-5}$
$x = \pm\sqrt {(-1)(5)}$
$x = \pm i\sqrt {5}$
Second factor:
$x + 2 = 0$
$x = -2$
The solutions are $-2, -i\sqrt {5}, \text{ and } -i\sqrt {5}$.
