The sum of a natural number and its square is 156. Find the number.

Explanation

To solve the problem, we need to find a natural number \( x \) such that the sum of the number and its square equals 156. We can set up the equation based on the given information.

### Step-by-step Solution:

1. **Set Up the Equation**:
We are given that the sum of a natural number \( x \) and its square \( x^2 \) is equal to 156. We can express this as:
\[ x + x^2 = 156 \]

2. **Rearrange the Equation**:
To form a standard quadratic equation, we can rearrange the equation:
\[ x^2 + x - 156 = 0 \]

3. **Factor the Quadratic Equation**:
We need to factor the quadratic equation \( x^2 + x - 156 = 0 \). We are looking for two numbers that multiply to \(-156\) and add to \(1\) (the coefficient of \(x\)). The numbers \(13\) and \(-12\) satisfy these conditions:
\[ (x + 13)(x - 12) = 0 \]

4. **Solve for \( x \)**:
Now we can set each factor equal to zero:
\[ x + 13 = 0 \quad \text{or} \quad x - 12 = 0 \]
Solving these gives:
\[ x = -13 \quad \text{or} \quad x = 12 \]

5. **Determine the Natural Number**:
Since we are looking for a natural number, we discard \( x = -13 \) because natural numbers are positive integers. Therefore, the solution is:
\[ x = 12 \]

### Final Answer:
The natural number is \( \boxed{12} \).