Solving the Cubic Equation: (t4)(3t+1)(t+2)=0
This equation represents a cubic polynomial, meaning it has a highest power of 3. Solving for t involves finding the values that make the entire equation equal to zero.
Utilizing the Zero Product Property
The key to solving this equation is the Zero Product Property, which states: If the product of two or more factors is zero, then at least one of the factors must be zero.
In our case, we have three factors: (t4), (3t+1), and (t+2). To make the entire equation equal to zero, at least one of these factors needs to be zero.
Finding the Solutions
Let's solve for t in each factor:

(t4) = 0
 Adding 4 to both sides: t = 4

(3t+1) = 0
 Subtracting 1 from both sides: 3t = 1
 Dividing both sides by 3: t = 1/3

(t+2) = 0
 Subtracting 2 from both sides: t = 2
Conclusion
Therefore, the solutions to the cubic equation (t4)(3t+1)(t+2)=0 are:
 t = 4
 t = 1/3
 t = 2
These are the values of t that make the equation true. They represent the roots of the cubic polynomial.