The problem asks us to find the sum of $BA + AE$, given that:

1. $AB \perp BC$, meaning angle $\angle ABC = 90^\circ$,
2. Points $D$ and $E$ are on lines $AB$ and $BC$ respectively,
3. $BA + AE = BD + DC$,
4. The lengths $AD = 2$, $BE = 3$, and $EC = 4$ are given.

Let's break this down:

Step-by-Step Solution:

We are asked to find $BA + AE$. From the given condition $BA + AE = BD + DC$, we can work to express the lengths of $BD$ and $DC$ in terms of the known values.

1. Labeling the segments:

   • Let $BA = x$, so $AE = x + 2$ (since $AD = 2$).
   • Therefore, $BD = BE + AE = 3 + (x + 2) = x + 5$.
   • We know $EC = 4$, so $DC = 4 + AE = 4 + (x + 2) = x + 6$.

2. Using the equation $BA + AE = BD + DC$:
   From the equation, we have: $BA + AE = BD + DC$ Substituting in the expressions: $x + (x + 2) = (x + 5) + (x + 6)$ Simplifying both sides: $2x + 2 = 2x + 11$ The terms involving $x$ cancel out, leaving: $2 = 11,$ which is clearly a contradiction. Therefore, there is likely an issue with the problem setup or diagram, or further context may be needed.

Would you like further clarifications or details on how to approach this differently?