Factorise Fraction Fb1Df4

1. **State the problem:**\n\nFactorise the expression $$\frac{4x - 3 + 2(3 - x)}{2x^2 - x - 6}$$.\n\n2. **Simplify the numerator:**\n\nExpand the term $2(3 - x)$:\n$$4x - 3 + 2(3 - x) = 4x - 3 + 6 - 2x = (4x - 2x) + (-3 + 6) = 2x + 3$$\n\n3. **Factorise the denominator:**\n\nThe denominator is a quadratic: $$2x^2 - x - 6$$\nWe look for two numbers that multiply to $2 \times (-6) = -12$ and add to $-1$. These numbers are $-4$ and $3$.\n\nRewrite the middle term using these numbers:\n$$2x^2 - 4x + 3x - 6$$\nGroup terms:\n$$ (2x^2 - 4x) + (3x - 6)$$\nFactor each group:\n$$2x(x - 2) + 3(x - 2)$$\nFactor out the common binomial:\n$$ (2x + 3)(x - 2)$$\n\n4. **Rewrite the expression with factored numerator and denominator:**\n$$\frac{2x + 3}{(2x + 3)(x - 2)}$$\n\n5. **Simplify the fraction by canceling common factors:**\n$$\frac{\cancel{2x + 3}}{\cancel{2x + 3}(x - 2)} = \frac{1}{x - 2}$$\n\n6. **Final answer:**\n\n$$\boxed{\frac{1}{x - 2}}$$\n\nThis is the simplified form of the original expression after factorisation and cancellation.