X+11+ Vx+11 = 42

To solve the equation \( x + 11 + \sqrt{x + 11} = 42 \), we first isolate the square root term.

1. Subtract \( x + 11 \) from both sides:

   \[ \sqrt{x + 11} = 42 - (x + 11) \]

   Simplifying the right side gives:

   \[ \sqrt{x + 11} = 31 - x \]

2. Now, square both sides to eliminate the square root:

   \[ (\sqrt{x + 11})^2 = (31 - x)^2 \]

   This simplifies to:

   \[ x + 11 = (31 - x)(31 - x) \]

   Expanding the right-hand side:

   \[ x + 11 = 961 - 62x + x^2 \]

3. Rearranging the equation gives us:

   \[ 0 = x^2 - 63x + 950 \]

4. Now, we can solve this quadratic equation using the quadratic formula:

   \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

   Here, \( a = 1 \), \( b = -63 \), and \( c = 950 \).

   \[ b^2 - 4ac = (-63)^2 - 4(1)(950) = 3969 - 3800 = 169 \]

5. Now substituting back into the quadratic formula:

   \[ x = \frac{63 \pm \sqrt{169}}{2} \]

   Since \( \sqrt{169} = 13 \):

   \[ x = \frac{63 \pm 13}{2} \]

6. This gives us two solutions:

   \[ x = \frac{76}{2} = 38 \]

   and

   \[ x = \frac{50}{2} = 25 \]

7. We need to check both solutions in the original equation to ensure they are valid:

   For \( x = 38 \): \[ 38 + 11 + \sqrt{38 + 11} = 49 + 7 = 56 \quad (\text{not valid}) \]

   For \( x = 25 \): \[ 25 + 11 + \sqrt{25 + 11} = 36 + 6 = 42 \quad (\text{valid}) \]

Thus, the solution to the equation \( x + 11 + \sqrt{x + 11} = 42 \) is

\[ \boxed{25} \]