Final answer:
The least common denominator for the given expression is (x+4)(x+2)(x-3), and the domain is all real numbers except x = -4, x = -2, and x = 3.
Step-by-step explanation:
To determine the least common denominator (LCD) for the expression 2x-3/x²+6+8 + 10/x²+x-12, we need to factor the denominators and find the common factors. The denominators are x²+6+8 and x²+x-12. The factors of x²+6+8 are (x+4)(x+2) and the factors of x²+x-12 are (x+4)(x-3). The common factors are (x+4) and (x+2)(x-3). Therefore, the LCD is (x+4)(x+2)(x-3).
The domain of the problem is the set of all real numbers except for the values that make the denominators equal to zero. To find these values, set each denominator equal to zero and solve for x. The values that make x²+6+8 equal to zero are x = -4 and x = -2. The values that make x²+x-12 equal to zero are x = 3 and x = -4. Therefore, the domain of the problem is all real numbers except x = -4, x = -2, and x = 3.