Final answer:
By comparing the given expression 10x² + 6y² - 16xy with the expanded form of 2(x - y)(ax + by), we find that a = 5. The value of b seems to be incorrect as per the given options, indicating a possible typo in the question since the correct value for b is -3, which is not listed as an option.
Step-by-step explanation:
The expression 10x² + 6y² - 16xy can be factored into the form 2(x - y)(ax + by). To find the values of a and b, we need to expand the factored form and compare it to the original expression. Starting by expanding, we get:
- 2(ax² + bxy - ayx - by²)
- 2(ax² - ayx + bxy - by²)
- 2(ax² - (a - b)xy - by²)
Comparing this with the original expression, we can see that:
- The coefficient of x² in the original expression is 10, meaning 2a must equal 10, so a = 5.
- The coefficient of y² in the original expression is 6, meaning -2b must equal 6, so b = -3.
- There seems to be a mistake with 'b' since we expect a positive value. On closer inspection, it becomes clear that there is a sign error in the factored form. The correct form should be 2(x - y)(5x - 3y) where a = 5 and b = -3, but since negative value is not presented in the options, and considering the proper factorisation form should be positive, there might be a typo.
By comparing the coefficients, we determine that a must equal 5, which corresponds to option (d), but 'b' does not match. However, since we have a sign error, the correct matching option for the values of a and b where both are positive is not provided in the question. If 'b' were -3, the correct combination would be (a = 5, b = -3), but this is not an option given.