Let’s solve this step-by-step.

Problem Breakdown:

1. Given:
   • $BC = 5$
   • $BE = 3$ (where $E$ is a point on $CD$)
   • $D$ is the foot of the perpendicular from $A$ to $BC$.
   • We need to compute:
     $AB^2 - AE^2 + 6CD$

Step 1: Use the Pythagorean theorem in $\triangle ABD$:

Let:

• $AB = x$
• $AD = h$ (perpendicular height from $A$ to $BC$)
• $BD = d$.

From the Pythagorean theorem in $\triangle ABD$:
$AB^2 = AD^2 + BD^2$ Thus: $AB^2 = h^2 + d^2$

Step 2: Relationship in $\triangle BCD$:

In $\triangle BCD$, we know $BC = 5$, and $D$ lies on $BC$. Assume: $BD = d \quad \text{and} \quad CD = 5 - d.$

Step 3: Use $BE = 3$:

Since $E$ lies on $CD$, we find $CE$ in $\triangle BCE$. Using the Pythagorean theorem:
$BE^2 = BC^2 - CE^2$ $3^2 = 5^2 - CE^2$ $CE^2 = 25 - 9 = 16 \quad \Rightarrow \quad CE = 4.$

Thus, $E$ divides $CD$ such that:
$CD = CE + ED = 4 + (5 - 4) = 5.$

Step 4: Compute $AE^2$:

From $\triangle AEC$, $AE^2$ involves height $AD$. Using $AD = h$ and $CD = 5$:
$AE^2 = AD^2 + DE^2 = h^2 + (5 - 4)^2 = h^2 + 1.$

Step 5: Substitute values into the expression:

The required expression is:
$AB^2 - AE^2 + 6CD$

Substitute:
$AB^2 = h^2 + d^2, \quad AE^2 = h^2 + 1, \quad CD = 5.$ $AB^2 - AE^2 + 6CD = (h^2 + d^2) - (h^2 + 1) + 6(5).$ Simplify:
$AB^2 - AE^2 + 6CD = d^2 - 1 + 30 = d^2 + 29.$

Step 6: Use $d = 3$:

From $BE = 3$, $d = 3$:
$AB^2 - AE^2 + 6CD = 3^2 + 29 = 9 + 29 = 38.$

Final Answer:

The computed value does not match the multiple-choice options. Let me recheck.I found a potential oversight in the calculation. Let’s carefully re-evaluate each step and ensure we get an answer matching the choices.

Re-evaluation:

1. Given Data:

   • $BC = 5$, $BE = 3$, $AB^2 - AE^2 + 6CD$ is to be calculated.
   • $D$ is the foot of the perpendicular from $A$ to $BC$, and $CD = 5 - d$, where $BD = d$.

2. Triangle BCE: From earlier work, $CE = 4$, and $CD = 5$.

3. Recalculate $AB^2 - AE^2$: Let’s write: $AB^2 = h^2 + BD^2 \quad \text{and} \quad AE^2 = h^2 + (5 - 4)^2 = h^2 + 1.$ Then: $AB^2 - AE^2 = (h^2 + BD^2) - (h^2 + 1) = BD^2 - 1.$

4. Substitute $BD = 3$: Since $BD = BE = 3$: $AB^2 - AE^2 = 3^2 - 1 = 9 - 1 = 8.$

5. Add $6CD$: $CD = 5$, so: $6CD = 6 \cdot 5 = 30.$

6. Final Calculation: Combine terms: $AB^2 - AE^2 + 6CD = 8 + 30 = 38.$

Correct Answer:

The result still does not match the options. This discrepancy might involve an error in the problem setup or interpretation. Could you clarify the setup or ask for additional insights?