14g4 + 67g2a6 + 18a12 = ?

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(2g2+9a6)(8g2+2a6)
(2g2-9a6)(7g2-2a6)
(2g2+9a6)(7g2+3a6)
(2g2+9a6)(7g2+2a6)

To solve the given equation:

\[
14g^4 + 67g^2a^6 + 18a^{12}
\]

we need to factor it in a way that matches the options provided.

Let's rewrite and examine it step by step.

1. **Identify the structure of the polynomial**:  
   Notice the given polynomial is a quadratic in form:
   \[
   (14g^4) + (67g^2a^6) + (18a^{12})
   \]

   This looks like a quadratic in terms of \( g^2 \) and \( a^6 \).

2. **Compare it with the possible factorized forms**:
   - Consider the polynomial as \( Ax^2 + Bxy + Cy^2 \) where \( x = g^2 \) and \( y = a^6 \).

3. **Check the options**:
   We need to find which factorization gives us the original polynomial:
   
   Let's expand each option and see which one matches.

- **Option 1: (2g² + 9a⁶)(8g² + 2a⁶)**

  Expand it:
  \[
  (2g^2)(8g^2) + (2g^2)(2a^6) + (9a^6)(8g^2) + (9a^6)(2a^6)
  \]
  \[
  = 16g^4 + 4g^2a^6 + 72g^2a^6 + 18a^{12}
  \]
  \[
  = 16g^4 + 76g^2a^6 + 18a^{12}
  \]
  This does not match.

- **Option 2: (2g² - 9a⁶)(7g² - 2a⁶)**

  Expand it:
  \[
  (2g^2)(7g^2) + (2g^2)(-2a^6) + (-9a^6)(7g^2) + (-9a^6)(-2a^6)
  \]
  \[
  = 14g^4 - 4g^2a^6 - 63g^2a^6 + 18a^{12}
  \]
  \[
  = 14g^4 - 67g^2a^6 + 18a^{12}
  \]
  This does not match.

- **Option 3: (2g² + 9a⁶)(7g² + 3a⁶)**

  Expand it:
  \[
  (2g^2)(7g^2) + (2g^2)(3a^6) + (9a^6)(7g^2) + (9a^6)(3a^6)
  \]
  \[
  = 14g^4 + 6g^2a^6 + 63g^2a^6 + 27a^{12}
  \]
  \[
  = 14g^4 + 69g^2a^6 + 27a^{12}
  \]
  This does not match.

- **Option 4: (2g² + 9a⁶)(7g² + 2a⁶)**

  Expand it:
  \[
  (2g^2)(7g^2) + (2g^2)(2a^6) + (9a^6)(7g^2) + (9a^6)(2a^6)
  \]
  \[
  = 14g^4 + 4g^2a^6 + 63g^2a^6 + 18a^{12}
  \]
  \[
  = 14g^4 + 67g^2a^6 + 18a^{12}
  \]

This matches exactly with the original expression. Therefore, the correct answer is:

**(2g² + 9a⁶)(7g² + 2a⁶)**

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Would you like further details or have any questions?

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Here are some related questions for practice:

1. What are the general steps to factorize quadratic polynomials?
2. How can you identify the correct pair of binomials that factor a trinomial?
3. What is the process for expanding binomials to check for factorization?
4. How do you recognize a quadratic form in variables with exponents?
5. What are some examples of polynomials that can be factored by grouping?
6. How does changing the coefficients in a trinomial affect its factorization?
7. What is the difference between factorization and simplification of polynomials?
8. How do you solve a polynomial equation after it has been factorized?

**Tip**: Always double-check your expansions when factoring polynomials, as small errors can lead to incorrect factorization.

Explore how to factorize the polynomial 14g^4 + 67g^2a^6 + 18a^12 step-by-step with detailed explanations. Learn about polynomial factorization techniques and apply them to quadratic forms involving variables with exponents. Suitable for high school students.

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