1. Factorize $p^2 - 8p + 16$. Step 1: Recognize this as a quadratic trinomial. Step 2: Check if it is a perfect square: $p^2 - 8p + 16 = (p - 4)^2$. 2. Factorize $m^2 + 6m - 27$. Step 1: Find two numbers that multiply to $-27$ and add to $6$. Step 2: These numbers are $9$ and $-3$. Step 3: Factor as $(m + 9)(m - 3)$. 3. Factorize $m^2 - 3m - 10$. Step 1: Find two numbers that multiply to $-10$ and add to $-3$. Step 2: These numbers are $-5$ and $2$. Step 3: Factor as $(m - 5)(m + 2)$. 4. Factorize $p^2 - 3p + 2$. Step 1: Find two numbers that multiply to $2$ and add to $-3$. Step 2: These numbers are $-1$ and $-2$. Step 3: Factor as $(p - 1)(p - 2)$. 5. Factorize $4y^3 + 12y^2 - 72y$. Step 1: Factor out the greatest common factor $4y$. Step 2: $4y(y^2 + 3y - 18)$. Step 3: Factor quadratic: find numbers multiplying to $-18$ and adding to $3$ are $6$ and $-3$. Step 4: Final factorization: $4y(y + 6)(y - 3)$. 6. Factorize $25b^2 + 35b - 30$. Step 1: Factor out the greatest common factor $5$. Step 2: $5(5b^2 + 7b - 6)$. Step 3: Factor quadratic: numbers multiplying to $-30$ and adding to $7$ are $10$ and $-3$. Step 4: $5(5b - 3)(b + 2)$. 7. Factorize $20x^2 + 100x + 125$. Step 1: Factor out the greatest common factor $5$. Step 2: $5(4x^2 + 20x + 25)$. Step 3: Recognize perfect square: $(2x + 5)^2$. Step 4: Final factorization: $5(2x + 5)^2$. 8. Factorize $20n^2 - 100n + 125$. Step 1: Factor out the greatest common factor $5$. Step 2: $5(4n^2 - 20n + 25)$. Step 3: Recognize perfect square: $(2n - 5)^2$. Step 4: Final factorization: $5(2n - 5)^2$. 9. Factorize $4p^2 - 20p + 25$. Step 1: Recognize perfect square: $(2p - 5)^2$. 10. Factorize $12 + 11x - x^2$. Step 1: Rewrite as $-x^2 + 11x + 12$. Step 2: Multiply by $-1$ to get $x^2 - 11x - 12$. Step 3: Find two numbers multiplying to $-12$ and adding to $-11$: $-12$ and $1$. Step 4: Factor as $(x - 12)(x + 1)$. Step 5: Reintroduce negative sign: $-(x - 12)(x + 1)$. 11. Factorize $x^2 - 4$. Step 1: Recognize difference of squares: $(x - 2)(x + 2)$. 12. Factorize $3p^2 - 27q^2$. Step 1: Factor out $3$: $3(p^2 - 9q^2)$. Step 2: Recognize difference of squares: $3(p - 3q)(p + 3q)$. 13. Factorize $4y^2 - 9$. Step 1: Recognize difference of squares: $(2y - 3)(2y + 3)$. 14. Factorize $a^2 b^2 - 1$. Step 1: Recognize difference of squares: $(ab - 1)(ab + 1)$. 15. Factorize $25x^2 + 16y^2$. Step 1: This is a sum of squares, which does not factor over the reals. 16. Factorize $12x^2 - 75$. Step 1: Factor out $3$: $3(4x^2 - 25)$. Step 2: Recognize difference of squares: $3(2x - 5)(2x + 5)$. 17. Factorize $169m^2 - 36u^2$. Step 1: Recognize difference of squares: $(13m - 6u)(13m + 6u)$. 18. Factorize $y^6 - 100$. Step 1: Recognize difference of squares: $(y^3 - 10)(y^3 + 10)$. 19. Factorize $4x^2y^2 - 9y^2$. Step 1: Factor out $y^2$: $y^2(4x^2 - 9)$. Step 2: Recognize difference of squares: $y^2(2x - 3)(2x + 3)$. 20. Factorize $400 - 4n^2$. Step 1: Factor out $4$: $4(100 - n^2)$. Step 2: Recognize difference of squares: $4(10 - n)(10 + n)$. 21. Simplify $\frac{2a - 4}{a^2 + a - 6}$. Step 1: Factor numerator: $2(a - 2)$. Step 2: Factor denominator: $(a + 3)(a - 2)$. Step 3: Cancel $(a - 2)$. Step 4: Result: $\frac{2}{a + 3}$. 22. Simplify $\frac{y^2 + 5y + 4}{y^2 - 4y - 5}$. Step 1: Factor numerator: $(y + 4)(y + 1)$. Step 2: Factor denominator: $(y - 5)(y + 1)$. Step 3: Cancel $(y + 1)$. Step 4: Result: $\frac{y + 4}{y - 5}$. 23. Simplify $\frac{x^2 - 25}{x^2 - 5x}$. Step 1: Factor numerator: $(x - 5)(x + 5)$. Step 2: Factor denominator: $x(x - 5)$. Step 3: Cancel $(x - 5)$. Step 4: Result: $\frac{x + 5}{x}$. 24. Simplify $\frac{12x^6}{27x^4}$. Step 1: Simplify coefficients: $\frac{12}{27} = \frac{4}{9}$. Step 2: Simplify powers: $x^{6-4} = x^2$. Step 3: Result: $\frac{4}{9}x^2$. 25. Simplify $\frac{100a^3 b}{150a b^3}$. Step 1: Simplify coefficients: $\frac{100}{150} = \frac{2}{3}$. Step 2: Simplify variables: $a^{3-1} = a^2$, $b^{1-3} = b^{-2} = \frac{1}{b^2}$. Step 3: Result: $\frac{2}{3} \frac{a^2}{b^2}$. 26. Simplify $\frac{18 a^3 b^3 c^4}{9 a^7 b^2 c^2}$. Step 1: Simplify coefficients: $\frac{18}{9} = 2$. Step 2: Simplify variables: $a^{3-7} = a^{-4} = \frac{1}{a^4}$, $b^{3-2} = b$, $c^{4-2} = c^2$. Step 3: Result: $2 \frac{b c^2}{a^4}$. 27. Simplify $\frac{3x + 18}{x^2 + 6x}$. Step 1: Factor numerator: $3(x + 6)$. Step 2: Factor denominator: $x(x + 6)$. Step 3: Cancel $(x + 6)$. Step 4: Result: $\frac{3}{x}$. 28. Simplify $\frac{6x + 24}{x^2 - 7x + 12}$. Step 1: Factor numerator: $6(x + 4)$. Step 2: Factor denominator: $(x - 3)(x - 4)$. Step 3: No common factors to cancel. Step 4: Result: $\frac{6(x + 4)}{(x - 3)(x - 4)}$. 29. Simplify $\frac{x^2 - 5x + 6}{x^2 + 2x - 15}$. Step 1: Factor numerator: $(x - 2)(x - 3)$. Step 2: Factor denominator: $(x + 5)(x - 3)$. Step 3: Cancel $(x - 3)$. Step 4: Result: $\frac{x - 2}{x + 5}$. 30. Simplify $\frac{P^2 + pq}{P^2 - pr}$. Step 1: Factor numerator: $P^2 + pq$ cannot be factored further. Step 2: Factor denominator: $P^2 - pr$ cannot be factored further. Step 3: No common factors. Step 4: Result: $\frac{P^2 + pq}{P^2 - pr}$. Final answers are provided with detailed steps for each problem.