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The equation provided is:

\[
y^3 - 4y + 15 = 0
\]

This is a cubic equation in \(y\). To solve it, we generally look for rational roots using the Rational Root Theorem, and if that doesn’t yield results, numerical or graphical methods like factoring or using synthetic division may help.

### Steps:

1. **Check for Rational Roots**:
   By the Rational Root Theorem, possible rational roots could be factors of the constant term (15) divided by the leading coefficient (1). The possible roots are:
   \[
   \pm 1, \pm 3, \pm 5, \pm 15
   \]
   Let's try substituting these into the equation to see if any of them satisfy it.

2. **Substituting \(y = -3\)**:
   \[
   (-3)^3 - 4(-3) + 15 = -27 + 12 + 15 = 0
   \]
   So, \(y = -3\) is a root.

3. **Factor the Equation**:
   Since \(y = -3\) is a root, we can factor \(y^3 - 4y + 15\) as:
   \[
   (y + 3)(y^2 - 3y + 5) = 0
   \]

4. **Solve the Quadratic**:
   Now solve the quadratic \(y^2 - 3y + 5 = 0\) using the quadratic formula:
   \[
   y = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(5)}}{2(1)} = \frac{3 \pm \sqrt{9 - 20}}{2} = \frac{3 \pm \sqrt{-11}}{2}
   \]
   Since \(\sqrt{-11}\) involves imaginary numbers, the other solutions are:
   \[
   y = \frac{3 \pm i\sqrt{11}}{2}
   \]

### Final Solutions:
The equation \(y^3 - 4y + 15 = 0\) has the following solutions:
1. \( y = -3 \)
2. \( y = \frac{3 + i\sqrt{11}}{2} \)
3. \( y = \frac{3 - i\sqrt{11}}{2} \)

Would you like further details on any of these steps or have any additional questions?

Here are 5 related questions to expand understanding:
1. How does the Rational Root Theorem help in solving cubic equations?
2. What is synthetic division, and how does it assist in factoring cubic polynomials?
3. What is the significance of imaginary numbers in quadratic solutions?
4. Can cubic equations always be factored, or are there exceptions?
5. How can we graphically interpret the solutions of a cubic equation?

**Tip:** If a polynomial has a complex solution, its conjugate is also a solution, as seen here with \(\frac{3 + i\sqrt{11}}{2}\) and \(\frac{3 - i\sqrt{11}}{2}\).

Explore how to solve the cubic equation y^3 - 4y + 15 = 0 step-by-step. This solution uses the Rational Root Theorem and the quadratic formula, and includes imaginary number concepts. Suitable for high school students.

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