Question

1. multiply \\(\frac{x^2 + 5x - 14}{x^2 - 5x + 6} \cdot \frac{x^2 + 2x - 15}{x^2 + 12x + 35}\\). completely simplify your answer and state any variable restrictions.\

\\(0\\); the variable restrictions are \\(x \
eq -7, x \
eq -5, x \
eq 2, x \
eq 3\\)\
\\(x\\); the variable restrictions are \\(x \
eq -7, x \
eq -5, x \
eq 2, x \
eq 3\\)\
\\(1\\); the variable restrictions are \\(x \
eq -7, x \
eq -5, x \
eq 2, x \
eq 3\\)\
\\(x^2\\); the variable restrictions are \\(x \
eq -7, x \
eq -5, x \
eq 2, x \
eq 3\\)

Explanation:

Step1: Factor numerators/denominators

First fraction: \(\frac{x^2+5x-14}{x^2-5x+6} = \frac{(x+7)(x-2)}{(x-2)(x-3)}\)
Second fraction: \(\frac{x^2+2x-15}{x^2+12x+35} = \frac{(x+5)(x-3)}{(x+7)(x+5)}\)

Step2: Multiply fractions

\(\frac{(x+7)(x-2)(x+5)(x-3)}{(x-2)(x-3)(x+7)(x+5)}\)

Step3: Cancel common factors

All pairs cancel, result = 1
Restrictions: Denominators ≠0 → \(x≠-7,-5,2,3\)

Answer:

1; the variable restrictions are \(x
eq -7, x
eq -5, x
eq 2, x
eq 3\)