SOLUTION: solve for x (3x-2)(x+5)=4x+2

x (3x-2)(x+5)=4x+2

Let's open up one of the brackets first

x(3x^2 + 15x - 2x - 10) = 4x + 2

Let's open up the last bracket

3x^3 + 15x^2 - 2x^2 - 10x = 4x + 2

Now let's move all the variables to the left hand side

3x^3 + 13x^2 -14x - 2 = 0 

Now we have a cubic equation.

To solve this, we use the factor theorem.

f(x) = 3x^3 + 13x^2 -14x - 2

When f(a) = 0, then (x-a) is a factor of f(x)

f(0) = -2

f(-1) = -3 + 13 + 14 -2 = 22

f(1) = 3 + 13 -14 -2 = 0

f(1) = 0, therefore (x-1) is a factor of f(x)

We now need to divide f(x) by (x-1) and find a quadratic equation.

I have 'magically' divided it. (using the long division method)

f(x) = (3x^2 + 16x + 2)(x-1)

We now have a quadratic and a linear.

(3x^2 + 16x + 2)(x-1)  = 0

Solve linear:

x-1 = 0
x = 1 --> First solution


Solve quadratic:

3x^2 + 16x + 2 = 0

 
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation  (in our case ) has the following solutons:
  
  
  
  For these solutions to exist, the discriminant  should not be a negative number.
  
  First, we need to compute the discriminant : .
  
  Discriminant d=232 is greater than zero. That means that there are two solutions: .
  
    
    
    
    Quadratic expression  can be factored:
  
  Again, the answer is: -0.128075631378697, -5.20525770195464.
Here's your graph:





We now know our 3 solutions!

x= 1, -5.21, -0.128