how to solve this equation in the real number system:

Question

2x^(3) + 3x^(2) + 2x + 3 = 0

steps please

Answer 1

2X^3 + 3X^2 + 2X + 3 = 0.

First, we reduce the cubic Eq to a guadratic Eq by finding a value of x that satisfies the Eq. This was done by
trial and error. However, the task was made a lot easier by the use of EXCEL Spread sheets.

It was determined x = -1.5 satisfies the cubic Eq:

x = -1.5 = -1 1/2 = -3/2.

x = -3/2,
x + 3/2 = 0,
Multiply both sides by 2 and get:
2x + 3 = 0.
Using long division, we divide the cubic Eq by 2x + 3:

(2x^3 + 3x^2 +2x + 3)/(2x + 3)=
x^2 + 1.
Now we can easily solve the quad. Eq:

x^2 + 1 = 0,

x2 = -1,
x = sqrt(-1) = +- i.

Solution set: x = - 3/2, x = i, x = -i.

So there are 3 solutions which is the max. for a cubic Eq.