Answer:
see below
Step-by-step explanation:
This is solved by simplifying each expression and identifying the column heading it matches.
1. (6 x^2/(x^2 - 7 x + 10)) / (2 x)/(x - 5))
= (6 x^2/(x^2 - 7 x + 10)) · (x -5)/(2 x) . . . . invert and multiply
= (6x^2)/(2x) · (x -5)/((x -2)(x -5)) . . . . . . . factor so common factors can cancel
= 3x/(x -2) . . . . matches column 1
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2. (x - 4) (x + 2)/(x^2 + 5 x + 6) + (-3 x^2 + 24 x - 20)/((x + 3) (4 x - 5))
= (x -4)(x +2)/((x +3)(x +2)) + (-3x^2 +24x -20)/((x +3)(4x -5)) . . . factor
= (x -4)/(x +3) + (-3x^2 +24x -20)/((x +3)(4x -5)) . . . . cancel common factor
= ((x -4)(4x -5) +(-3x^2 +24x -20))/((x +3)(4x -5)) . . . . use common denominator
= (4x^2 -21x +20 -3x^2 +24x -20)/((x +3)(4x -5)) . . . expand product
= (x^2 +3x)/((x +3)(4x -5)) . . . . . . collect terms
= (x)(x +3)/((x +3)(4x -5)) . . . . . . factor numerator
= x/(4x -5) . . . . . . . . . . . . . . . . . . cancel common factor ... matches column 2
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3. 3 x^2/(x + 3) · (2 x + 6)/(2 x^2 - 4 x)
= 3·2·x·(x +3)/((x +3)(2·x)(x -2)) = 3/(x -2) . . . . matches column 1
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4. 3 x/(4 x - 5) - 4 x^2/(8 x^2 - 10 x)
= 3x/(4x -5) - 4x^2/(2x(4x -5)) = (3x -2x)/(4x -5) = x/(4x -5) ... matches col 2
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5. 5 x^2/(x - 2) · (2 x + 6)/(8 x^2 - 4 x) . . . . no match (has a denominator factor of 2x-1 that doesn't cancel any numerator factors)
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6. -x/(4 x - 5) - 4 x^2/(16 x^2 - 22 x)
In order to match one of the columns, the term on the right must reduce to 2x/(4x-5), which it does not, or must have a denominator factor of x-2, which it also does not. no match.
