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The equation to solve by factoring is:

\[
12w^2 - 8w = -3w + 25
\]

### Step 1: Rearrange the equation
First, move all terms to one side of the equation so that the right-hand side equals zero:

\[
12w^2 - 8w + 3w - 25 = 0
\]

Simplify the terms:

\[
12w^2 - 5w - 25 = 0
\]

### Step 2: Factor the quadratic equation
We need to factor \(12w^2 - 5w - 25 = 0\). Using the AC method, multiply \(12 \times (-25) = -300\). Now, we look for two numbers that multiply to give \(-300\) and add to \(-5\). The numbers are \(-20\) and \(15\).

Rewriting the middle term:

\[
12w^2 - 20w + 15w - 25 = 0
\]

Now, group the terms:

\[
(12w^2 - 20w) + (15w - 25) = 0
\]

Factor each group:

\[
4w(3w - 5) + 5(3w - 5) = 0
\]

Factor out the common factor \((3w - 5)\):

\[
(4w + 5)(3w - 5) = 0
\]

### Step 3: Solve for \(w\)
Now, solve each factor:

1. \(4w + 5 = 0\)
   \[
   4w = -5 \quad \Rightarrow \quad w = -\frac{5}{4}
   \]

2. \(3w - 5 = 0\)
   \[
   3w = 5 \quad \Rightarrow \quad w = \frac{5}{3}
   \]

### Step 4: Verify the solution
The solutions are \(w = -\frac{5}{4}\) and \(w = \frac{5}{3}\).

### Answer
The correct option is:

\[
\boxed{w = -\frac{5}{4}, \, w = \frac{5}{3}}
\]

Would you like more details or have any other questions?

### Related Questions:
1. How do you factor quadratics when the leading coefficient isn't 1?
2. What is the AC method in factoring?
3. How do you solve quadratic equations by completing the square?
4. How can you use the quadratic formula to solve \(12w^2 - 5w - 25 = 0\)?
5. What are the different methods to solve quadratic equations?

### Tip:
When factoring, always check for a greatest common factor (GCF) before applying more advanced techniques!

Solve 12w^2 - 8w = -3w + 25 by factoring.

Learn how to solve the quadratic equation 12w^2 - 8w + 3w - 25 = 0 by factoring. This step-by-step guide covers algebraic concepts like factoring and the zero product property. Suitable for students in Grades 9-12.

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