Certainly! Let's solve the equation step-by-step:
Given the equation: [tex]\( 5u(u + 2)(u - 6) = 0 \)[/tex].
### Step 1: Identify the roots of the equation using the zero-product property.
According to the zero-product property, if a product of multiple factors is equal to zero, then at least one of the factors must be zero. Hence, we set each factor equal to zero and solve for [tex]\( u \)[/tex].
Factor 1: [tex]\( 5u \)[/tex]
[tex]\[
5u = 0 \implies u = 0
\][/tex]
Factor 2: [tex]\( u + 2 \)[/tex]
[tex]\[
u + 2 = 0 \implies u = -2
\][/tex]
Factor 3: [tex]\( u - 6 \)[/tex]
[tex]\[
u - 6 = 0 \implies u = 6
\][/tex]
### Step 2: List all the solutions.
The solutions to the equation [tex]\( 5u(u + 2)(u - 6) = 0 \)[/tex] are:
[tex]\[
u = -2, \, u = 0, \, u = 6
\][/tex]
Therefore, the complete set of solutions is:
[tex]\[
\boxed{-2, \, 0, \, 6}
\][/tex]
