To solve the quadratic equation \( 2x^2 + 12x + 3 = 15 \) by factoring, follow these steps:

### Step 1: Move all terms to one side of the equation
We first rewrite the equation in standard form by moving the 15 to the left-hand side:

\[
2x^2 + 12x + 3 - 15 = 0
\]

This simplifies to:

\[
2x^2 + 12x - 12 = 0
\]

### Step 2: Factor out the greatest common factor (GCF)
Notice that 2 is a common factor of all terms, so factor out 2:

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

### Step 3: Factor the quadratic expression (if possible)
Now, we need to factor \( x^2 + 6x - 6 \). Unfortunately, this quadratic doesn't factor nicely into rational numbers. So, we can use the **quadratic formula** to solve for \( x \).

### Step 4: Use the quadratic formula
The quadratic formula is:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \( x^2 + 6x - 6 = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = 6 \)
- \( c = -6 \)

Substituting these values into the quadratic formula:

\[
x = \frac{-6 \pm \sqrt{6^2 - 4(1)(-6)}}{2(1)}
\]
\[
x = \frac{-6 \pm \sqrt{36 + 24}}{2}
\]
\[
x = \frac{-6 \pm \sqrt{60}}{2}
\]
\[
x = \frac{-6 \pm \sqrt{60}}{2}
\]

Since \( \sqrt{60} \) simplifies to \( 2\sqrt{15} \), we get:

\[
x = \frac{-6 \pm 2\sqrt{15}}{2}
\]

Simplifying further:

\[
x = -3 \pm \sqrt{15}
\]

### Final Solution:
The solutions are:

\[
x = -3 + \sqrt{15} \quad \text{or} \quad x = -3 - \sqrt{15}
\]

Would you like further details or have any questions about this process?

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Here are 5 related questions:
1. How do you simplify a quadratic expression by factoring?
2. What are the steps for completing the square?
3. How do you use the quadratic formula for different types of quadratic equations?
4. When is factoring not possible, and what methods can be used instead?
5. Can all quadratic equations be solved by factoring?

**Tip**: Always check if there is a common factor before attempting more complex solving methods!

Learn how to solve the quadratic equation 2x^2 + 12x + 3 = 15 by factoring and using the quadratic formula. This problem involves algebraic concepts such as factoring and solving quadratics, making it suitable for students in Grades 9-10.

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