Question

select the correct answer.
consider functions f and g.
f(x) = \frac{x + 12}{x^2 + 4x - 12}, for x \
eq 2 and x \
eq -6
g(x) = \frac{4x^2 - 16x + 16}{4x + 48}, for x \
eq -12
which expression is equal to f(x) \cdot g(x) ?
\bigcirc a. \frac{x + 2}{x - 6}
\bigcirc b. \frac{4x - 8}{x + 6}
\bigcirc c. \frac{1}{x + 6}
\bigcirc d. \frac{x - 2}{x + 6}

Explanation:

Step1: Factor denominators and numerators

Factor \(x^2 + 4x - 12\): \(x^2 + 4x - 12=(x + 6)(x - 2)\)
Factor \(4x^2 - 16x + 16\): \(4x^2 - 16x + 16 = 4(x^2 - 4x + 4)=4(x - 2)^2\)
Factor \(4x + 48\): \(4x + 48 = 4(x + 12)\)

Step2: Substitute factored forms into \(f(x)\) and \(g(x)\)

\(f(x)=\frac{x + 12}{(x + 6)(x - 2)}\)
\(g(x)=\frac{4(x - 2)^2}{4(x + 12)}\)

Step3: Multiply \(f(x)\) and \(g(x)\)

\(f(x)\cdot g(x)=\frac{x + 12}{(x + 6)(x - 2)}\cdot\frac{4(x - 2)^2}{4(x + 12)}\)

Step4: Simplify the product

Cancel common factors: \(x + 12\) cancels, \(4\) cancels, \((x - 2)\) cancels once.
Result: \(\frac{(x - 2)}{(x + 6)}\)

Answer:

D. \(\frac{x - 2}{x + 6}\)