Find The discriminant of the equation `(x+1)^3=4-x+x^3`

Answer

To find the discriminant of the equation \((x+1)^3 = 4 - x + x^3\), we will follow these steps:

### Step 1: Expand the left-hand side
We start with the equation:
\[ (x + 1)^3 = 4 - x + x^3 \]
Using the binomial expansion for \((x + 1)^3\):
\[ (x + 1)^3 = x^3 + 3x^2 + 3x + 1 \]
So, we can rewrite the equation as:
\[ x^3 + 3x^2 + 3x + 1 = 4 - x + x^3 \]

### Step 2: Simplify the equation
Next, we can cancel \(x^3\) from both sides:
\[ 3x^2 + 3x + 1 = 4 - x \]
Now, we will move all terms to one side of the equation:
\[ 3x^2 + 3x + x + 1 - 4 = 0 \]
This simplifies to:
\[ 3x^2 + 4x - 3 = 0 \]

### Step 3: Identify coefficients
Now we have a quadratic equation in the standard form \(ax^2 + bx + c = 0\):
\[ 3x^2 + 4x - 3 = 0 \]
Here, we identify:
- \(a = 3\)
- \(b = 4\)
- \(c = -3\)

### Step 4: Calculate the discriminant
The discriminant \(D\) of a quadratic equation is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ D = 4^2 - 4 \cdot 3 \cdot (-3) \]
Calculating this gives:
\[ D = 16 + 36 = 52 \]

### Final Answer
Thus, the discriminant of the equation is:
\[ \boxed{52} \]