To factorize the quadratic expression \( 6x^2 + x - 15 \), we will use the method of **factor by grouping** or **trial and error**. We are looking for two binomials of the form \( (ax + b)(cx + d) \).

### Step 1: Identify coefficients
The quadratic is in the form \( ax^2 + bx + c \), where:
- \( a = 6 \)
- \( b = 1 \)
- \( c = -15 \)

### Step 2: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( a \times c = 6 \times -15 = -90 \), and add to \( b = 1 \).

The two numbers are \( 10 \) and \( -9 \), because:
- \( 10 \times -9 = -90 \)
- \( 10 + (-9) = 1 \)

### Step 3: Rewrite the middle term
We rewrite the quadratic expression by splitting the middle term:
\[
6x^2 + 10x - 9x - 15
\]

### Step 4: Factor by grouping
Now, group the terms:
\[
(6x^2 + 10x) - (9x + 15)
\]

Factor out the greatest common factor from each group:
\[
2x(3x + 5) - 3(3x + 5)
\]

### Step 5: Factor out the common binomial
Now, factor out the common binomial \( (3x + 5) \):
\[
(3x + 5)(2x - 3)
\]

### Final Answer:
The factorization of \( 6x^2 + x - 15 \) is:
\[
(3x + 5)(2x - 3)
\]

Would you like more details or have any further questions?

Here are some related questions:
1. How can you solve a quadratic equation using factoring?
2. What other methods can be used to factor quadratics besides grouping?
3. How can you determine if a quadratic is factorable?
4. What are some real-world problems modeled by quadratic equations?
5. How does factoring relate to solving quadratic equations by the quadratic formula?

**Tip:** When factoring quadratics, always check if the leading coefficient (the number in front of \( x^2 \)) is greater than 1, as this can make factoring more challenging.

Learn how to factorize the quadratic expression 6x^2 + x - 15 using the method of factor by grouping. Detailed step-by-step solution with key algebraic concepts and formulas. Suitable for students in Grades 9-12.

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