Final answer:
Brute-force factorization tests all possible factors, while Brent-Pollard and Williams-Lenstra use specific sequences and elliptic curves respectively. Quadratic sieve factorization uses quadratic equations and modular arithmetic.
Step-by-step explanation:
Brute-force factorization: In this method, all possible factors of a given number are tested exhaustively one by one until the prime factors are identified.
Brent-Pollard factorization: This is an improvement over brute-force method where a sequence of numbers is generated and tested for factors.
Williams-Lenstra factorization: This method is based on the concept of elliptic curves and uses them to find the prime factors of a number.
Quadratic sieve factorization: This advanced method uses quadratic equations and modular arithmetic to find the prime factors of a large number.