To solve the quadratic equation \( 6x^2 + 5x - 4 = 0 \) by factoring, follow these steps:

### Step 1: Multiply the leading coefficient and the constant term
Multiply the coefficient of \( x^2 \) (which is 6) by the constant term (which is -4):

\[
6 \times -4 = -24
\]

### Step 2: Find two numbers that multiply to -24 and add to the middle coefficient
We need to find two numbers that multiply to \(-24\) and add to \(5\) (the coefficient of \(x\)).

These numbers are \(8\) and \(-3\):

\[
8 \times -3 = -24 \quad \text{and} \quad 8 + (-3) = 5
\]

### Step 3: Rewrite the middle term using these two numbers
Rewrite the equation by splitting the middle term (5x) using the two numbers we found:

\[
6x^2 + 8x - 3x - 4 = 0
\]

### Step 4: Factor by grouping
Group the terms to factor them:

\[
(6x^2 + 8x) - (3x + 4) = 0
\]

Now, factor out the common factors in each group:

\[
2x(3x + 4) - 1(3x + 4) = 0
\]

### Step 5: Factor out the common binomial factor
Now, factor out the common binomial factor \((3x + 4)\):

\[
(2x - 1)(3x + 4) = 0
\]

### Step 6: Solve for \(x\)
Set each factor equal to zero and solve for \(x\):

1. \(2x - 1 = 0\)

   \[
   2x = 1 \quad \Rightarrow \quad x = \frac{1}{2}
   \]

2. \(3x + 4 = 0\)

   \[
   3x = -4 \quad \Rightarrow \quad x = -\frac{4}{3}
   \]

### Final Answer:
The solutions to the equation are:

\[
x = \frac{1}{2} \quad \text{or} \quad x = -\frac{4}{3}
\]

Would you like more details or have any questions? Here are some related questions:

1. How can you check your solutions for a quadratic equation?
2. What if the quadratic equation cannot be factored easily?
3. How does the quadratic formula work as an alternative method?
4. What is the significance of the discriminant in a quadratic equation?
5. Can quadratic equations always be factored into real number factors?

**Tip:** When factoring, always look for the greatest common factor (GCF) first before applying other methods.

Learn how to solve the quadratic equation 6x^2 + 5x - 4 using factoring with a detailed step-by-step solution. This problem covers key algebraic concepts, including factoring, quadratic equations, and the quadratic formula theorem. Suitable for students in Grades 9-12.

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