Question

multiply the rational expressions. give your answer in factored form.\\(\frac{k^{2}+k - 6}{k^{2}+15k + 56}cdot\frac{k^{2}+2k - 48}{k^{2}-4k + 4}\\)

Explanation:

Step1: Factor all quadratic polynomials

1. $k^2 + k - 6 = (k+3)(k-2)$
2. $k^2 + 15k + 56 = (k+7)(k+8)$
3. $k^2 + 2k - 48 = (k+8)(k-6)$
4. $k^2 - 4k + 4 = (k-2)^2$

Step2: Substitute factors into expression

$\frac{(k+3)(k-2)}{(k+7)(k+8)} \cdot \frac{(k+8)(k-6)}{(k-2)^2}$

Step3: Cancel common factors

Cancel $(k-2)$ and $(k+8)$ from numerator/denominator:
$\frac{(k+3)}{(k+7)} \cdot \frac{(k-6)}{(k-2)}$

Step4: Multiply remaining terms

$\frac{(k+3)(k-6)}{(k+7)(k-2)}$

Answer:

$\frac{(k+3)(k-6)}{(k+7)(k-2)}$