Final answer:
To multiply the given fractions, factor each polynomial, cancel out the common factors, and find the simplified expression. The expression simplifies to (x - 3)/(x + 2), with restrictions that x cannot be -5, -1, 5, or -2, as these values would make the original denominators zero.
Step-by-step explanation:
The question asks to multiply two fractions and find any restrictions on the variables. The fractions are:
((x^2-25)/(x^2+6x+5)) * ((x^2-2x-3)/(x^2-3x-10)).
To multiply these fractions, you can first factor each polynomial:
- x^2 - 25 = (x + 5)(x - 5)
- x^2 + 6x + 5 = (x + 5)(x + 1)
- x^2 - 2x - 3 = (x - 3)(x + 1)
- x^2 - 3x - 10 = (x - 5)(x + 2)
Multiplying the factored forms, you get:
((x + 5)(x - 5))/((x + 5)(x + 1)) * ((x - 3)(x + 1))/((x - 5)(x + 2)).
Simplifying the expression by canceling out common factors, we are left with:
(x - 3)/(x + 2).
The restrictions on the variable x are that x cannot take values that make the denominator of any original fraction equal to zero. Therefore, x cannot be -5, -1, 5, or -2.