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Let's solve the given quadratic trinomial \( 6x^2 - 13x - 5 \) by factoring.

### Step 1: Identify the coefficients
We have the quadratic in the form \( ax^2 + bx + c \):
- \( a = 6 \)
- \( b = -13 \)
- \( c = -5 \)

### Step 2: Multiply \( a \) and \( c \)
We multiply the coefficients \( a \) and \( c \):  
\( 6 \times (-5) = -30 \)

### Step 3: Find two numbers that multiply to give \( -30 \) and add to give \( -13 \)
We need two numbers whose product is \( -30 \) and whose sum is \( -13 \). These numbers are \( -15 \) and \( 2 \), since:
- \( -15 \times 2 = -30 \)
- \( -15 + 2 = -13 \)

### Step 4: Rewrite the middle term
We rewrite \( -13x \) as \( -15x + 2x \):
\[
6x^2 - 15x + 2x - 5
\]

### Step 5: Factor by grouping
Group the terms in pairs:
\[
(6x^2 - 15x) + (2x - 5)
\]
Now factor each group:
\[
3x(2x - 5) + 1(2x - 5)
\]

### Step 6: Factor out the common binomial
Both terms contain \( (2x - 5) \), so factor that out:
\[
(3x + 1)(2x - 5)
\]

Thus, the factored form of \( 6x^2 - 13x - 5 \) is:
\[
(3x + 1)(2x - 5)
\]

Let me know if you need more details or have any questions.

Here are 5 related questions to practice:
1. Factor the trinomial \( 8x^2 - 14x - 4 \).
2. Solve the equation \( 3x^2 + 7x - 10 = 0 \) by factoring.
3. Factor \( 12x^2 - 7x - 10 \).
4. How do you factor \( x^2 - 4x - 12 \)?
5. Factor \( 5x^2 + 16x + 3 \).

**Tip:** When factoring, always check if you can factor out a greatest common factor (GCF) before proceeding with other methods.

Learn how to factor the trinomial 6x² - 13x - 5 step-by-step. This solution explains factoring by grouping and covers key algebra concepts for students in Grades 9-10.

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