To solve the equation \( x + (x + 2) + (x + 4) = 120 \), we first simplify the left side:
\[ x + x + 2 + x + 4 = 120 \]
Combining like terms, we get:
\[ 3x + 6 = 120 \]
Next, we subtract 6 from both sides:
\[ 3x = 120 - 6 \] \[ 3x = 114 \]
Now, we divide both sides by 3:
\[ x = \frac{114}{3} = 38 \]
Now that we have the first integer \( x \), we can find the other two integers:
1st integer: \( x = 38 \) 2nd integer: \( x + 2 = 38 + 2 = 40 \) 3rd integer: \( x + 4 = 38 + 4 = 42 \)
Thus, the three consecutive even integers are 38, 40, and 42. The largest of the three integers is:
\[ \boxed{42} \]