Understanding the Fourth Proportional
The fourth proportional to three quantities \( a, b, c \) is defined as the quantity \( d \) such that the ratio of \( a \) to \( b \) is equal to the ratio of \( c \) to \( d \). Mathematically, it can be expressed as:
\[ \frac{a}{b} = \frac{c}{d} \]
From this, we can derive that:
\[ d = \frac{b \cdot c}{a} \]
In this case, the quantities given are \( x, 2x, (x + 1) \).

Identifying the Variables
- \( a = x \)
- \( b = 2x \)
- \( c = x + 1 \)

Finding the Fourth Proportional
Using the formula for the fourth proportional, we plug in the identified variables:
\[ d = \frac{b \cdot c}{a} \]
Substituting the values:
\[ d = \frac{(2x) \cdot (x + 1)}{x} \]

Calculating the Expression
1. **Multiply** \( 2x \) and \( (x + 1) \):
- \( 2x(x + 1) = 2x^2 + 2x \)
2. **Divide by** \( x \):
- \( d = \frac{2x^2 + 2x}{x} \)
- Simplifying gives:
\[ d = 2x + 2 \]

Conclusion
The fourth proportional to \( x, 2x, (x + 1) \) is \( 2x + 2 \). This demonstrates the relationship between the quantities and provides a clear method to find the fourth proportional in a straightforward manner.