The equation given is $y = 8x^2 + 24x + 15$, and you are asked to determine which of the given options (A, B, C, D) represents the same function, using the technique of completing the square.

Completing the Square:

1. Start with the quadratic equation: $y = 8x^2 + 24x + 15$

2. Factor out the $8$ from the terms involving $x$: $y = 8(x^2 + 3x) + 15$

3. To complete the square inside the parentheses, take half of the coefficient of $x$ (which is 3), square it, and add and subtract that value inside the parentheses: $y = 8\left( x^2 + 3x + \left(\frac{3}{2}\right)^2 - \left(\frac{3}{2}\right)^2 \right) + 15$ Simplify: $y = 8\left( (x + \frac{3}{2})^2 - \frac{9}{4} \right) + 15$

4. Distribute the $8$ and simplify: $y = 8(x + \frac{3}{2})^2 - 18 + 15$ $y = 8(x + \frac{3}{2})^2 - 3$

Thus, the equivalent equation is: $y = 8\left( x + \frac{3}{2} \right)^2 - 3$

Conclusion:

The correct answer is Option B: $y = 8\left( x + \frac{3}{2} \right)^2 - 3$

Would you like more details on any step? Here are 5 follow-up questions:

1. How do you factor out a coefficient when completing the square?
2. Why do we add and subtract the same number inside the parentheses?
3. Can you explain how to simplify after completing the square?
4. How does completing the square relate to transforming quadratic functions?
5. Why is distributing the constant after completing the square important?

Tip: Completing the square is a powerful method for rewriting quadratic equations and is often used for solving or finding the vertex form of the equation!