how many solution do this problem have?

3x(4x-5)(3x+7)(x-6)=0

how many variable factors do you see?

Each different variable factor yields a solution.

e.g the second factor of (4x-5) would give a solution of x = 5/4
etc

The degree of the equation is 4 (that is, if multplied, you would have a factor x^4. The number of roots is equal to the degree.

To determine the number of solutions to the equation 3x(4x-5)(3x+7)(x-6)=0, we first need to find the values of x that make the equation true.

To solve this equation, we can set each factor equal to zero individually and solve for x.

Setting the first factor equal to zero: 3x = 0.
Dividing both sides by 3, we find x = 0 as a solution.

Next, setting the second factor equal to zero: (4x-5) = 0.
Adding 5 to both sides and then dividing by 4, we find x = 5/4 as a solution.

Moving on to the third factor: (3x+7) = 0.
Subtracting 7 from both sides and then dividing by 3, we find x = -7/3 as a solution.

Lastly, setting the fourth factor equal to zero: (x-6) = 0.
Adding 6 to both sides, we find x = 6 as a solution.

Hence, the equation has four solutions: x = 0, x = 5/4, x = -7/3, and x = 6.