Final answer:
The factored form of the expression 13y^2 - 5yx - 8x^2 is (y - x)(13y + 8x). This is achieved by first finding two numbers that multiply to -104 and add to -5, which are -13 and 8, and then by grouping and factoring out the common factors.
Step-by-step explanation:
The question involves the task of factoring a quadratic expression completely. The given expression is 13y2 - 5yx - 8x2. To factorize this, it is often useful to look for two numbers that multiply to give the product of the coefficient of y2 and x2 (which is 13* -8 = -104) and add up to the coefficient of yx (which is -5).
Upon examination, the two numbers that fit this criterion are -13 and +8. Now, we can rewrite the middle term using these two numbers: 13y2 - 13yx + 8yx - 8x2. Then we group the terms and factor by grouping: (13y2 - 13yx) + (8yx - 8x2) = 13y(y - x) + 8x(y - x). Since both terms contain a common factor of (y - x), we can factor it out to get the completely factored expression: (y - x)(13y + 8x). Always remember to check your answer and ensure that it is reasonable by expanding the factors to see if you get the original expression.