Answer
The solutions are $x = 0$, $x = -5$, and $x = -2$.
Work Step by Step
First, we factor out what is common in both terms:
$x(x^2 + 7x + 10) = 0$
To factor this equation, we want to find which factors, when multiplied, will give us the product of the $a$ and $c$ terms but when added together will give us the $b$ term.
Let's look at possible factors:
$5$ and $2$
$10$ and $1$
The first combination will work.
Let's write the equation in factor form:
$x(x + 5)(x + 2) = 0$
According to the zero product property, if the product of factors equals $0$, then each of the factors can be $0$; therefore, we can set each of these factors equal to $0$ and solve.
Let's set the first factor equal to $0$:
$x = 0$
For the second factor, we subtract $5$ from each side of the equation:
$x = -5$
For the third factor, we subtract $2$ from each side of the equation:
$x = -2$
The solutions are $x = 0$, $x = -5$, and $x = -2$.
To check our answers, we substitute each of these values into the original equation to see if the two sides equal one another.
Let's look at the first factor:
$(0)^3 + 7(0)^2 + 10(0) = 0$
Evaluate exponents first:
$0 + 7(0) + 10(0) = 0$
Multiply to simplify:
$0 + 0 - 0 = 0$
Add first two terms:
$0 - 0 = 0$
Subtract:
$0 = 0$
The two sides of the equation are equal to one another; therefore, this solution is correct. Let's check the second solution:
$(-5)^3 + 7(-5)^2 + 10(-5) = 0$
Evaluate exponents first:
$-125 + 7(25) + 10(-5) = 0$
Multiply to simplify:
$-125 + 175 - 50 = 0$
Add first two terms:
$50 - 50 = 0$
Subtract:
$0 = 0$
The two sides of the equation are equal to one another; therefore, this solution is correct. Let's check the third solution:
$(-2)^3 + 7(-2)^2 + 10(-2) = 0$
Evaluate exponents first:
$-8 + 7(4) + 10(-2) = 0$
Multiply to simplify:
$-8 + 28 - 20 = 0$
Add first two terms:
$20 - 20 = 0$
Subtract:
$0 = 0$
The two sides of the equation are equal to one another; therefore, this solution is correct.
