The function h(x) = 5x2 – x – 6 is the result of combining two functions using multiplication.

Which functions could have been combined? 

The function \( h(x) = 5x^2 - x - 6 \) can be expressed as the product of two linear functions. Let's find those functions.

We start by factoring the quadratic function \( h(x) = 5x^2 - x - 6 \).

1. **Factor the quadratic expression:**

   We are looking for two numbers that multiply to \( 5 \times (-6) = -30 \) and add up to \(-1\) (the coefficient of \( x \)).

   These two numbers are \( -6 \) and \( 5 \).

2. **Rewrite the middle term:**

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

3. **Group and factor by grouping:**

   \[
   (5x^2 - 6x) + (5x - 6)
   \]

   Factor out the common factors in each group:

   \[
   x(5x - 6) + 1(5x - 6)
   \]

   This gives us:

   \[
   (x + 1)(5x - 6)
   \]

So, the function \( h(x) = 5x^2 - x - 6 \) can be expressed as the product of the functions \( f(x) = x + 1 \) and \( g(x) = 5x - 6 \).

**Answer:**

The two functions that could have been combined using multiplication are:

\[
f(x) = x + 1 \quad \text{and} \quad g(x) = 5x - 6
\]

Would you like more details on how the factorization was done, or do you have any questions?

Here are some related questions you might find interesting:
1. How can we factor a quadratic function that cannot be easily factored using integers?
2. What is the relationship between the roots of a quadratic function and its factors?
3. How can we verify that two functions are factors of a given polynomial?
4. How do we use the quadratic formula to solve for roots when factorization is difficult?
5. How does multiplying linear functions result in a quadratic function?

**Tip:** When factoring quadratics, always look for pairs of factors that multiply to the product of the leading coefficient and the constant term, and add to the middle coefficient.

Explore how to factorize the quadratic function 5x^2 - x - 6 into two linear functions. This problem involves understanding quadratic equations and their factorization using algebraic techniques. Suitable for high school students in Grades 9-12.

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