1. **State the problem:** Find the partial fraction decomposition of the rational expression $$\frac{10}{5x^2 - 11x + 2}$$. 2. **Factor the denominator:** We need to factor the quadratic polynomial in the denominator. $$5x^2 - 11x + 2$$ We look for two numbers that multiply to $$5 \times 2 = 10$$ and add to $$-11$$. These numbers are $$-10$$ and $$-1$$. Rewrite the middle term: $$5x^2 - 10x - x + 2$$ Group terms: $$5x(x - 2) - 1(x - 2)$$ Factor out the common binomial: $$(5x - 1)(x - 2)$$ 3. **Set up the partial fractions:** Since the denominator factors as $$(5x - 1)(x - 2)$$, the decomposition is: $$\frac{10}{(5x - 1)(x - 2)} = \frac{A}{5x - 1} + \frac{B}{x - 2}$$ where $$A$$ and $$B$$ are constants to be determined. 4. **Multiply both sides by the denominator to clear fractions:** $$10 = A(x - 2) + B(5x - 1)$$ 5. **Expand the right side:** $$10 = A x - 2A + 5 B x - B$$ Group like terms: $$10 = (A + 5B) x + (-2A - B)$$ 6. **Equate coefficients of like terms:** For $$x$$ term: $$A + 5B = 0$$ For constant term: $$-2A - B = 10$$ 7. **Solve the system of equations:** From $$A + 5B = 0$$, we get $$A = -5B$$. Substitute into the second equation: $$-2(-5B) - B = 10$$ $$10B - B = 10$$ $$9B = 10$$ $$B = \frac{10}{9}$$ Then, $$A = -5 \times \frac{10}{9} = -\frac{50}{9}$$ 8. **Write the final partial fraction decomposition:** $$\frac{10}{5x^2 - 11x + 2} = \frac{-\frac{50}{9}}{5x - 1} + \frac{\frac{10}{9}}{x - 2} = -\frac{50}{9(5x - 1)} + \frac{10}{9(x - 2)}$$ This is the partial fraction decomposition with integer and fractional coefficients as required.