1. **State the problem:** Simplify the expression $$6 - \frac{x + 5}{(7x - 5)(x + 4)}$$. 2. **Identify the formula and rules:** To simplify, we need to combine terms by finding a common denominator. The first term 6 can be written as $$\frac{6(7x - 5)(x + 4)}{(7x - 5)(x + 4)}$$ to have the same denominator. 3. **Rewrite the expression:** $$6 - \frac{x + 5}{(7x - 5)(x + 4)} = \frac{6(7x - 5)(x + 4)}{(7x - 5)(x + 4)} - \frac{x + 5}{(7x - 5)(x + 4)}$$ 4. **Expand the numerator of the first fraction:** $$6(7x - 5)(x + 4) = 6[(7x)(x) + (7x)(4) - 5(x) - 5(4)] = 6(7x^2 + 28x - 5x - 20) = 6(7x^2 + 23x - 20)$$ 5. **Multiply out:** $$6(7x^2 + 23x - 20) = 42x^2 + 138x - 120$$ 6. **Combine the fractions:** $$\frac{42x^2 + 138x - 120}{(7x - 5)(x + 4)} - \frac{x + 5}{(7x - 5)(x + 4)} = \frac{42x^2 + 138x - 120 - (x + 5)}{(7x - 5)(x + 4)}$$ 7. **Simplify the numerator:** $$42x^2 + 138x - 120 - x - 5 = 42x^2 + 137x - 125$$ 8. **Final simplified expression:** $$\frac{42x^2 + 137x - 125}{(7x - 5)(x + 4)}$$ 9. **Check for factorization:** Try to factor numerator $42x^2 + 137x - 125$. 10. **Use the quadratic formula to check roots:** $$x = \frac{-137 \pm \sqrt{137^2 - 4 \times 42 \times (-125)}}{2 \times 42}$$ Calculate discriminant: $$137^2 = 18769$$ $$4 \times 42 \times 125 = 21000$$ $$18769 + 21000 = 39769$$ $$\sqrt{39769} = 199.42 \text{ (not a perfect square)}$$ Since the discriminant is not a perfect square, numerator does not factor nicely. **Answer:** $$6 - \frac{x + 5}{(7x - 5)(x + 4)} = \frac{42x^2 + 137x - 125}{(7x - 5)(x + 4)}$$