We are given the equation:

\[
2x^4 - 2x^3 - 7x^2 - 7x + 14 = 0
\]

### Step 1: Factor out the greatest common factor (GCF)
The GCF of all the terms is 1, so we can't factor out anything other than 1.

### Step 2: Try factoring by grouping
Let’s attempt to group terms:

\[
(2x^4 - 2x^3) - (7x^2 + 7x) + 14 = 0
\]

Factor out the common terms within each group:

\[
2x^3(x - 1) - 7x(x + 1) + 14 = 0
\]

However, this does not immediately lend itself to further simple factoring. 

### Step 3: Apply numerical methods or substitution
At this point, solving the equation either involves trial-and-error methods (testing rational roots like \( x = 1, -1, 2, -2 \)) or using numerical methods like the Rational Root Theorem or graphing methods. We can also apply a substitution-based method or solve it numerically for approximate values.

Would you like to explore one of these methods in more detail or see a numerical approximation for the roots?

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**Relative questions:**
1. Would you like to use the Rational Root Theorem for further factoring?
2. Do you prefer finding the roots using a numerical approximation?
3. Should we proceed with solving by substitution methods, like \( u = x^2 \)?
4. Would you like to try solving by graphing or another visual method?
5. Do you want a more detailed step-by-step breakdown for solving quartic equations?

**Tip**: Quartic equations can sometimes be solved by special factorization techniques like substitution, but in many cases, numerical approximations or graphing give quick insights into the solution!

Learn how to solve the quartic equation 2x^4 - 2x^3 - 7x^2 - 7x + 14 using factoring techniques, the Rational Root Theorem, and substitution methods. Explore detailed steps suitable for high school students in Grades 10-12.

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