To solve the expression \(6y^4 - 13y^2 - 5\), we can factor it, if possible. This expression is a quadratic in terms of \(y^2\). Let's rewrite it as:

\[
6(y^2)^2 - 13(y^2) - 5
\]

Now, let \(x = y^2\). The expression becomes:

\[
6x^2 - 13x - 5
\]

We need to factor this quadratic expression by finding two numbers that multiply to \(6 \times (-5) = -30\) and add up to \(-13\). Those numbers are \(-15\) and \(2\).

Using these factors, we can rewrite the expression:

\[
6x^2 - 15x + 2x - 5
\]

Next, factor by grouping:

\[
3x(2x - 5) + 1(2x - 5)
\]

\[
(3x + 1)(2x - 5)
\]

Now, substitute back \(x = y^2\):

\[
(3y^2 + 1)(2y^2 - 5)
\]

So, the factored form of \(6y^4 - 13y^2 - 5\) is:

\[
(3y^2 + 1)(2y^2 - 5)
\]

Would you like more details or have any questions about this solution?

Here are some related questions that might help further your understanding:

1. How can you factor other types of polynomials, such as cubic or quartic equations?
2. What are the methods to solve equations once they are factored?
3. How does the substitution method help in factoring complex expressions?
4. What are other techniques for factoring polynomials, like completing the square?
5. How can we verify the factors of a polynomial are correct?

**Tip:** Always check your factored form by multiplying the factors to ensure they produce the original expression.

Explore how to factor the polynomial 6y^4 - 13y^2 - 5 step-by-step using substitution and factoring by grouping. This detailed solution breaks down algebraic concepts for high school students (Grades 9-11).

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