To solve the equation \((1 + x) + (2 + x) + (3 + x) + \ldots + (99 + x) = 6345\), we need to simplify and solve for \(x\).

First, let's rewrite the left-hand side of the equation by separating the constants and the \(x\) terms:

\[
(1 + x) + (2 + x) + (3 + x) + \ldots + (99 + x)
\]

This can be rewritten as:

\[
(1 + 2 + 3 + \ldots + 99) + (x + x + x + \ldots + x)
\]

There are 99 terms in the sequence, so the sum of the \(x\) terms is \(99x\).

Next, we need to find the sum of the integers from 1 to 99. The sum of the first \(n\) integers is given by the formula:

\[
\frac{n(n + 1)}{2}
\]

For \(n = 99\):

\[
\frac{99 \cdot 100}{2} = 4950
\]

So, the equation becomes:

\[
4950 + 99x = 6345
\]

Now, solve for \(x\):

\[
99x = 6345 - 4950
\]

\[
99x = 1395
\]

\[
x = \frac{1395}{99}
\]

\[
x = 14.0909090909 \ldots
\]

Since the problem does not specify that \(x\) must be an integer, we can leave it in this form or simplify it further. The exact value is:

\[
x = \frac{465}{33} \approx 14.0909090909 \ldots
\]

So, the value of \(x\) is approximately \(14.09\).