Final answer:
The product in lowest terms is 3/2(x+2) and the values of x to exclude from the domain are -2, 4, and 7.
Step-by-step explanation:
The question asks for the simplified product of two rational expressions and the values of x that must be excluded from the domain of the expression. The provided expressions are -3x+21/-2x-4 divided by x²-16/x-7. To find the product, we first factor the expressions, if possible.
For the first expression, we can factor out a -3 in the numerator and a -2 in the denominator, resulting in 3(x-7)/2(x+2). The second expression is a difference of squares which can be factored to (x+4)(x-4) in the numerator and x-7 in the denominator. Since division by a fraction is the same as multiplication by its reciprocal, the division problem becomes multiplication:
3(x-7)/2(x+2) × (x-7)/(x+4)(x-4)
When we multiply these, we get:
3(x-7)(x-7)/2(x+2)(x-4)(x+4)
After canceling out the common terms (x-7), we simplify to:
3/2(x+2)
The excluded values of x are those for which the denominators in the original equation are zero. These values are x=-2, x=4, x=-4, and x=7. However, x=-4 does not appear in the simplified form, so we only need to exclude x=-2, x=4, and x=7 from the domain.