fully fectorise $$ \frac{x^{2}+1 x+12}{2 x^{2}-4 x-1} $$

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Alright, I need to fully factorize the given rational expression:

$$
\frac{x^{2} + 1x + 12}{2x^{2} - 4x - 1}
$$

First, let's simplify the numerator and the denominator separately to see if there are any common factors that can be canceled out.

**Numerator: $$ x^{2} + 1x + 12 $$**
- This is a quadratic expression. To factor it, I'll look for two numbers that multiply to $$ 12 $$ (the constant term) and add up to $$ 1 $$ (the coefficient of the middle term).
- The pairs of factors of $$ 12 $$ are:
  - $$ 1 \times 12 $$ (sum: $$ 13 $$)
  - $$ 2 \times 6 $$ (sum: $$ 8 $$)
  - $$ 3 \times 4 $$ (sum: $$ 7 $$)
- None of these pairs add up to $$ 1 $$, so the numerator doesn't factor nicely over the integers. It might be irreducible or require the quadratic formula.

**Denominator: $$ 2x^{2} - 4x - 1 $$**
- This is also a quadratic expression. To factor it, I'll look for two numbers that multiply to $$ (2 \times -1) = -2 $$ and add up to $$ -4 $$.
- The pairs of factors of $$ -2 $$ are:
  - $$ 1 \times -2 $$ (sum: $$ -1 $$)
  - $$ -1 \times 2 $$ (sum: $$ 1 $$)
- Neither of these pairs adds up to $$ -4 $$, so the denominator doesn't factor nicely over the integers either. It might require the quadratic formula.

**Attempting to Factor Both Numerator and Denominator:**
- Since both the numerator and the denominator don't factor easily, I'll consider factoring out the greatest common factor (GCF) if possible.
- For the numerator: $$ x^{2} + x + 12 $$, there's no GCF other than $$ 1 $$.
- For the denominator: $$ 2x^{2} - 4x - 1 $$, the GCF is $$ 1 $$.

**Checking for Common Factors:**
- There are no common factors between the numerator and the denominator that can be canceled out.

**Conclusion:**
- Since neither the numerator nor the denominator can be factored further over the integers, the expression cannot be simplified by factoring. Therefore, the fully factored form of the given rational expression is the expression itself.

The fully factored form of the expression is: $$ \frac{x^{2} + x + 12}{2x^{2} - 4x - 1} $$ This expression cannot be simplified further by factoring.

fully fectorise \( \frac{x^{2}+1 x+12}{2 x^{2}-4 x-1} \)

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