Final answer:
The division of the given algebraic expressions can be simplified by factorizing and canceling common factors, resulting in (x-5y)×5/(x-3y).
Step-by-step explanation:
To divide the given algebraic expressions, first we need to factorize both the numerator and the denominator where possible, and then simplify the expression by canceling out common factors.
The first part of the expression (x²-xy-20y²) can be factorized as (x-5y)(x+4y). Likewise, the second part of the denominator (x²-5xy+6y²) can be factorized as (x-2y)(x-3y).
To divide by ÷ (x+4y)/(5x-10y), we actually multiply by the reciprocal, which means multiplying by (5x-10y)/(x+4y). Notice that 5x-10y is 5(x-2y), which will cancel with one of the factors in the denominator of the first fraction.
So the division is as follows:
- (x-5y)(x+4y) ÷ (x-2y)(x-3y)
- × (5x-10y)/(x+4y)
- =(x-5y)(x+4y) × 5(x-2y) ÷ (x-2y)(x-3y)(x+4y)
- =(x-5y)×5 ÷ (x-3y)
After canceling out the common factors, the result is (x-5y)×5/(x-3y).