1. **State the problem:** Simplify the expression $$10p - \frac{8}{p^2 - 50p + 44}$$ and express it as a single fraction in simplest form. 2. **Factor the denominator:** The quadratic in the denominator is $$p^2 - 50p + 44$$. We look for factors of 44 that add up to -50. Since 44 is positive and the middle term is negative, both factors are negative. Factors of 44: 1, 2, 4, 11, 22, 44. Try -2 and -22: $$-2 + (-22) = -24$$ (not -50). Try -44 and -1: $$-44 + (-1) = -45$$ (not -50). Try -22 and -2: already tried. Try -11 and -4: $$-11 + (-4) = -15$$ (no). Try -1 and -44: no. Try -50 and something? No. Since no integer factors, use quadratic formula: $$p = \frac{50 \pm \sqrt{(-50)^2 - 4 \cdot 1 \cdot 44}}{2} = \frac{50 \pm \sqrt{2500 - 176}}{2} = \frac{50 \pm \sqrt{2324}}{2}$$ Since it does not factor nicely, keep denominator as is. 3. **Rewrite the expression as a single fraction:** Express $$10p$$ as $$\frac{10p(p^2 - 50p + 44)}{p^2 - 50p + 44}$$ to have common denominator: $$10p = \frac{10p(p^2 - 50p + 44)}{p^2 - 50p + 44}$$ 4. **Combine the fractions:** $$10p - \frac{8}{p^2 - 50p + 44} = \frac{10p(p^2 - 50p + 44) - 8}{p^2 - 50p + 44}$$ 5. **Expand the numerator:** $$10p(p^2 - 50p + 44) = 10p^3 - 500p^2 + 440p$$ So numerator is: $$10p^3 - 500p^2 + 440p - 8$$ 6. **Final expression:** $$\frac{10p^3 - 500p^2 + 440p - 8}{p^2 - 50p + 44}$$ 7. **Check for common factors:** Try to factor numerator: Look for common factor: none (10p^3, -500p^2, 440p, -8) share factor 2. Factor out 2: $$2(5p^3 - 250p^2 + 220p - 4)$$ No obvious factorization further. Denominator does not factor nicely. Therefore, the simplest form is: $$\boxed{\frac{10p^3 - 500p^2 + 440p - 8}{p^2 - 50p + 44}}$$