Question

multiply and simplify.
\\(\frac{a^{3}+a^{2}b + a + b}{3a^{3}+3a}\cdot\frac{6a^{2}}{3a^{2}-3b^{2}}\\)

\\(\frac{a^{3}+a^{2}b + a + b}{3a^{3}+3a}\cdot\frac{6a^{2}}{3a^{2}-3b^{2}}=\square\\)
(simplify your answer. type your answer in factored form.)

Explanation:

Step1: Factor numerators and denominators

Factor \(a^{3}+a^{2}b + a + b\): Group terms, \(a^{2}(a + b)+1(a + b)=(a^{2}+1)(a + b)\)
Factor \(3a^{3}+3a\): Factor out \(3a\), \(3a(a^{2}+1)\)
Factor \(3a^{2}-3b^{2}\): Factor out \(3\) and use difference of squares, \(3(a^{2}-b^{2}) = 3(a + b)(a - b)\)
Now the expression becomes:
\(\frac{(a^{2}+1)(a + b)}{3a(a^{2}+1)}\cdot\frac{6a^{2}}{3(a + b)(a - b)}\)

Step2: Cancel common factors

Cancel \((a^{2}+1)\) from numerator and denominator:
\(\frac{(a + b)}{3a}\cdot\frac{6a^{2}}{3(a + b)(a - b)}\)
Cancel \((a + b)\) from numerator and denominator:
\(\frac{1}{3a}\cdot\frac{6a^{2}}{3(a - b)}\)

Step3: Multiply and simplify

Multiply the numerators: \(1\times6a^{2}=6a^{2}\)
Multiply the denominators: \(3a\times3(a - b)=9a(a - b)\)
Simplify \(\frac{6a^{2}}{9a(a - b)}\): Cancel \(3a\) from numerator and denominator, \(\frac{2a}{3(a - b)}\)

Answer:

\(\frac{2a}{3(a - b)}\)