\[
\frac{a^3 + a + a^2 + 1}{a^3 + a^2 + ab^2 + b^2} \times \frac{2a^2 + 2ab + ab^2 + b^3}{a^3 + a + a^2b + b}
\]
Answer (2)
a 3 + a 2 + a b 2 + b 2 a 3 + a + a 2 + 1 ⋅ a 3 + a + a 2 b + b 2 a 2 + 2 ab + a b 2 + b 3 = a 2 ( a + 1 ) + b 2 ( a + 1 ) a ( a 2 + 1 ) + 1 ( a 2 + 1 ) ⋅ a ( a 2 + 1 ) + b ( a 2 + 1 ) 2 a ( a + b ) + b 2 ( a + b ) = = ( a + 1 ) ( a 2 + b 2 ) ( a 2 + 1 ) ( a + 1 ) ⋅ ( a 2 + 1 ) ( a + b ) ( a + b ) ( 2 a + b 2 ) = a 2 + b 2 2 a + b 2
The given algebraic expression can be simplified by factoring the numerators and denominators, followed by cancelling out common terms. The final simplified expression is a 2 + b 2 2 a + b 2 . This process demonstrates key algebraic techniques such as factorization and cancellation in rational expressions.
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