In how many points the line `y + 14 = 0` cuts the curve whose equation is `x(x^(2) + x + 1) + y = 0 ? `

Explanation

To solve the problem of finding how many points the line \( y + 14 = 0 \) cuts the curve defined by the equation \( x(x^2 + x + 1) + y = 0 \), we can follow these steps:

### Step 1: Rewrite the equations
The line can be rewritten as:
\[ y = -14 \]
The curve can be rewritten as:
\[ y = -x(x^2 + x + 1) \]

### Step 2: Set the two equations equal to each other
To find the points of intersection, we set the two expressions for \( y \) equal to each other:
\[ -14 = -x(x^2 + x + 1) \]
This simplifies to:
\[ x(x^2 + x + 1) = 14 \]

### Step 3: Rearrange the equation
Rearranging gives us:
\[ x^3 + x^2 + x - 14 = 0 \]

### Step 4: Use the Rational Root Theorem
We can use the Rational Root Theorem to test possible rational roots. Testing \( x = 2 \):
\[ 2^3 + 2^2 + 2 - 14 = 8 + 4 + 2 - 14 = 0 \]
Thus, \( x = 2 \) is a root.

### Step 5: Factor the polynomial
Now we can factor \( x^3 + x^2 + x - 14 \) using synthetic division or polynomial long division by \( x - 2 \):
\[ x^3 + x^2 + x - 14 = (x - 2)(x^2 + 3x + 7) \]

### Step 6: Analyze the quadratic factor
Next, we need to find the roots of the quadratic \( x^2 + 3x + 7 \). We can calculate the discriminant:
\[ D = b^2 - 4ac = 3^2 - 4 \cdot 1 \cdot 7 = 9 - 28 = -19 \]
Since the discriminant is negative, the quadratic has no real roots.

### Step 7: Conclusion
Thus, the only real solution to the original equation is \( x = 2 \). Therefore, the line \( y + 14 = 0 \) cuts the curve at only one point.

### Final Answer
The line cuts the curve at **1 point**.

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