Question

multiply.
\\(\dfrac{8x + 8}{6x + 30} \cdot \dfrac{2x + 10}{4x^2 - 4}\\)
\\(\dfrac{8x + 8}{6x + 30} \cdot \dfrac{2x + 10}{4x^2 - 4} = \square\\) (simplify your answer.)

Explanation:

Step1: Factor each expression

Factor numerators and denominators:

• \(8x + 8 = 8(x + 1)\)
• \(6x + 30 = 6(x + 5)\)
• \(2x + 10 = 2(x + 5)\)
• \(4x^2 - 4 = 4(x^2 - 1) = 4(x - 1)(x + 1)\) (using difference of squares: \(a^2 - b^2=(a - b)(a + b)\))

So the expression becomes:
\(\frac{8(x + 1)}{6(x + 5)} \cdot \frac{2(x + 5)}{4(x - 1)(x + 1)}\)

Step2: Cancel common factors

Cancel out common factors in numerators and denominators:

• Cancel \( (x + 1) \) from numerator and denominator.
• Cancel \( (x + 5) \) from numerator and denominator.
• Simplify coefficients: \(\frac{8}{6} \cdot \frac{2}{4}=\frac{8\times2}{6\times4}=\frac{16}{24}=\frac{2}{3}\)

After canceling, we have:
\(\frac{2}{3(x - 1)}\) (or \(\frac{2}{3x - 3}\))

Answer:

\(\frac{2}{3(x - 1)}\) (or \(\frac{2}{3x - 3}\))