Identify the greatest common factor of $$ 12 x^{2} $$ and $$ 10 b^{3} x^{4} $$

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To find the greatest common factor (GCF) of the expressions $$ 12 x^{2} $$ and $$ 10 b^{3} x^{4} $$, we will follow these steps:

1. **Factor the coefficients**: 
   - The coefficients are $$ 12 $$ and $$ 10 $$.
   - The prime factorization of $$ 12 $$ is $$ 2^2 \times 3 $$.
   - The prime factorization of $$ 10 $$ is $$ 2 \times 5 $$.

2. **Identify the common factors of the coefficients**:
   - The common prime factor is $$ 2 $$.
   - The minimum power of $$ 2 $$ in both factorizations is $$ 2^1 $$.

3. **Factor the variable parts**:
   - For $$ x^{2} $$ and $$ x^{4} $$, the common variable is $$ x $$.
   - The minimum power of $$ x $$ is $$ x^{2} $$.

4. **Combine the common factors**:
   - The GCF of the coefficients is $$ 2 $$.
   - The GCF of the variable parts is $$ x^{2} $$.

Thus, the GCF of $$ 12 x^{2} $$ and $$ 10 b^{3} x^{4} $$ is:

$$
2 x^{2}
$$

Therefore, the greatest common factor is $$ 2 x^{2} $$.
The greatest common factor is $$ 2x^{2} $$.

Identify the greatest common factor of \( 12 x^{2} \) and \( 10 b^{3} x^{4} \)

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