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Let's solve the problem step by step.

### Given:
1. The roots \( x_1 = 3 \) and \( x_2 = 5 \) of the equation:
   \[
   x^2 - (a + d)x + ad - bc = 0
   \]
2. The roots \( y_1 \) and \( y_2 \) of the equation:
   \[
   y^2 - (a^3 + d^3 + 3abc + 3bcd)y + (ad - bc)^3 = 0
   \]

We need to find the value of \( y_1^2y_2 + y_1y_2^2 \).

### Step 1: Use the sum and product of roots for the first equation.
For the quadratic equation \( x^2 - (a + d)x + ad - bc = 0 \):
- Sum of roots \( x_1 + x_2 = a + d \)
- Product of roots \( x_1x_2 = ad - bc \)

Given that \( x_1 = 3 \) and \( x_2 = 5 \):
- \( a + d = x_1 + x_2 = 3 + 5 = 8 \)
- \( ad - bc = x_1x_2 = 3 \times 5 = 15 \)

### Step 2: Use the sum and product of roots for the second equation.
For the quadratic equation \( y^2 - (a^3 + d^3 + 3abc + 3bcd)y + (ad - bc)^3 = 0 \):
- Sum of roots \( y_1 + y_2 = a^3 + d^3 + 3abc + 3bcd \)
- Product of roots \( y_1y_2 = (ad - bc)^3 \)

Given \( ad - bc = 15 \), we have:
- \( y_1y_2 = 15^3 = 3375 \)

### Step 3: Calculate \( y_1^2y_2 + y_1y_2^2 \).
We know:
\[
y_1^2y_2 + y_1y_2^2 = (y_1 + y_2) \times y_1y_2
\]
Thus:
\[
y_1^2y_2 + y_1y_2^2 = (a^3 + d^3 + 3abc + 3bcd) \times 3375
\]

To find \( y_1^2y_2 + y_1y_2^2 \), we need to calculate the expression for \( y_1 + y_2 \).

### Step 4: Simplify and calculate the result.
We know:
\[
y_1 + y_2 = a^3 + d^3 + 3abc + 3bcd
\]
Given \( a + d = 8 \), let's assume \( a = 3 \) and \( d = 5 \) (one possible solution):
- \( a^3 + d^3 = 3^3 + 5^3 = 27 + 125 = 152 \)
- \( 3abc + 3bcd \) requires the values of \( b \) and \( c \). However, the problem doesn't provide them directly, but it implies a symmetry in the problem, so it might suggest \( abc = bcd = 0 \) or some other simplification.

Let's simplify by using just the sum:
\[
y_1 + y_2 = 152
\]
Thus:
\[
y_1^2y_2 + y_1y_2^2 = 152 \times 3375 = 513000
\]

### Conclusion:
The correct answer is **D) 513000**.

Would you like more details, or do you have any questions?

### Related Questions:
1. How can we determine the values of \( b \) and \( c \) in such problems?
2. What happens if the roots \( x_1 \) and \( x_2 \) are different?
3. How would the equation change if \( ad - bc = 20 \)?
4. Can this method be applied to cubic equations as well?
5. What are the implications if \( y_1y_2 \) were negative?

### Tip:
In problems involving the sum and product of roots, carefully consider any symmetry or simplifications that might apply to simplify calculations.

Si las raíces de la ecuación x² - (a + d)x + ad - bc = 0 son x₁ = 3, x₂ = 5; y las raíces de la ecuación y² - (a³ + d³ + 3abc + 3bcd)y + (ad - bc)³ = 0 son y₁, y₂, Entonces el valor de y₁²y₂ + y₁y₂² es:

Learn how to solve a system of quadratic equations where the roots of x² - (a + d)x + ad - bc = 0 and y² - (a³ + d³ + 3abc + 3bcd)y + (ad - bc)³ = 0 are provided. This problem involves using the sum and product of roots formulas.

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