Answer:
c ∈ {-7, -3, 2, 5}
Step-by-step explanation:
"Illegal" values are those values that make the expression undefined—that is, they make the denominator zero.
The denominator of the product factors as (c+7)(c-2)(c-5)(c+3), so the values of c that will make this product zero are ones that make the factors zero:
... c = -7, +2, +5, -3
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Comment on factoring
Each of the quadratics is factored by finding two factors of the constant whose sum is the coefficient of the linear term. You want ...
... factors of -14 = -1·14 = -2·7 . . . . that add to give +5. They are -2 and 7, so the quadratic factors as (c-2)(c+7)
... factors of -15 = 1·(-15) = 3·(-5) . . . . that add to give -2. They are 3 and -5, so the quadratic factors as (c+3)(c-5).
