To factor the expression $y^3 + 125$, we recognize this as a sum of two cubes. The general formula for factoring a sum of two cubes $a^3 + b^3$ is:

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

For the given expression:

1. Rewrite $125$ as $5^3$.

   So, $y^3 + 125 = y^3 + (5)^3$.

2. Apply the sum of cubes formula with $a = y$ and $b = 5$:

$y^3 + 5^3 = (y + 5)(y^2 - 5y + 25)$

Final Factored Form:

$y^3 + 125 = (y + 5)(y^2 - 5y + 25)$

Would you like a breakdown of each step in more detail, or do you have any questions?

Here are some related questions to deepen your understanding:

1. How would you factor a difference of two cubes, such as $y^3 - 125$?
2. Can you factor $x^3 + 64$ using the sum of cubes formula?
3. What is the factored form of $8 + z^3$?
4. How does the formula for factoring sums of cubes differ from the difference of squares?
5. Can the sum of two squares, such as $y^2 + 25$, be factored using real numbers?

Tip: Always check if an expression can be rewritten as a sum or difference of cubes by identifying terms like $a^3$ or $b^3$ before applying the formula.