Final answer:
To simplify the expression, we need to find a common denominator for the two fractions and combine them. After expanding and simplifying the expression, we get (x^3 + 8x^2 + 3x - 42) / [(x - 2)(x + 5)(x + 6)(x + 5)].
Step-by-step explanation:
To simplify the expression, we need to find a common denominator for the two fractions. The denominators are both quadratic expressions, so we need to factor them to find the common denominator.
The first denominator, x^2 + 4x - 12, can be factored as (x - 2)(x + 6), and the second denominator, x^2 + 3x - 10, can be factored as (x - 2)(x + 5).
Therefore, the common denominator is (x - 2)(x + 6)(x + 5).
Now, we can multiply the numerators by the appropriate factors to get a common denominator.
The first numerator, x^2 + 6x + 5, can be multiplied by (x + 5) to get (x^2 + 6x + 5)(x + 5). The second numerator, 1, can be multiplied by (x + 6) to get (1)(x + 6).
Putting it all together, we have [(x^2 + 6x + 5)(x + 5)] / [(x - 2)(x + 6)(x + 5)] - [(1)(x + 6)] / [(x - 2)(x + 5)(x + 6)].
Now we can combine the fractions by finding a common denominator. Multiply the first fraction by (x - 2) / (x - 2) and the second fraction by (x + 5) / (x + 5) to get [(x^2 + 6x + 5)(x + 5)(x - 2)] / [(x - 2)(x + 6)(x + 5)(x + 5)] - [(1)(x + 6)(x - 2)] / [(x - 2)(x + 5)(x + 6)(x + 5)].
Simplifying further, we have [(x^2 + 6x + 5)(x + 5)(x - 2) - (x + 6)(x - 2)] / [(x - 2)(x + 5)(x + 6)(x + 5)].
Expanding and combining like terms in the numerator, we get [(x^3 + 10x^2 + 25x - 2x^2 - 12x - 30) - (x^2 - 8x - 12)] / [(x - 2)(x + 5)(x + 6)(x + 5)].
After simplifying the numerator, we have (x^3 + 8x^2 + 3x - 42) / [(x - 2)(x + 5)(x + 6)(x + 5)].
The complete question is: You and your classmates, Jack and Sian, are working to (x^(2)+6x+5)/(x^(2)+4x-12)-:(x^(2)+3x-10)/(x^(2)-36) is: