How to simplify an expression?I have tried to simplify this expression for quite a long time now but I can't find how to do it. ( 1 2 + 4 m − 1 − m 8 m 3 + 1 : 1 − 2 m 2 m 2 − 2 m + 1 ) ⋅ 4 m + 2 2 m − 1 − 1 1 − 4 m + 4 m 2 Can someone help me with it?

Question

How to simplify an expression?I have tried to simplify this expression for quite a long time now but I can&#039;t find how to do it.      (          1              2        +        4        m              −                  1        −        m                    8                  m          3                +        1              :                  1        −        2        m                    2                  m          2                −        2        m        +        1              )    ⋅            4      m      +      2              2      m      −      1        −      1          1      −      4      m      +      4              m        2            Can someone help me with it?

Shawn Castaneda · Accepted Answer

Starting with the original expression:      (          1              2        +        4        m              −                  1        −        m                    8                  m          3                +        1              :                  1        −        2        m                    2                  m          2                −        2        m        +        1              )    ⋅            4      m      +      2              2      m      −      1        −      1          1      −      4      m      +      4              m        2            Factoring where possible and changing the order of some terms:      (          1              2        (        2        m        +        1        )              +                  m        −        1                    (        2        m        +        1        )        (        4                  m          2                −        2        m        +        1        )              :                  −        (        2        m        −        1        )                    2                  m          2                −        2        m        +        1              )    ⋅            2      (      2      m      +      1      )              2      m      −      1        −      1          (      2      m      −      1              )        2            Converting division into multiplication by the reciprocal; distributing the outside fraction:      (          1              2        (        2        m        +        1        )                            2        (        2        m        +        1        )                    2        m        −        1              +                  m        −        1                    (        2        m        +        1        )        (        4                  m          2                −        2        m        +        1        )                            2                  m          2                −        2        m        +        1                    −        (        2        m        −        1        )                            2        (        2        m        +        1        )                    2        m        −        1              )    −      1          (      2      m      −      1              )        2            Some cancellations:        (          1              2        (        2        m        +        1        )                            2        (        2        m        +        1        )                    2        m        −        1              +                  m        −        1                              (          2          m          +          1          )                (        4                  m          2                −        2        m        +        1        )                            2                  m          2                −        2        m        +        1                    −        (        2        m        −        1        )                            2                  (          2          m          +          1          )                            2        m        −        1              )    −      1          (      2      m      −      1              )        2                ⟹        1          2      m      −      1        −            2      (      m      −      1      )              4              m        2            −      2      m      +      1                  2              m        2            −      2      m      +      1              (      2      m      −      1              )        2              −      1          (      2      m      −      1              )        2            Finding common denominators:            8              m        3            −      8              m        2            +      4      m      −      1              (      4              m        2            −      2      m      +      1      )      (      2      m      −      1              )        2              −            4              m        3            −      8              m        2            +      6      m      −      2              (      4              m        2            −      2      m      +      1      )      (      2      m      −      1              )        2              −            4              m        2            −      2      m      +      1              (      4              m        2            −      2      m      +      1      )      (      2      m      −      1              )        2            Adding/subtracting:            4              m        3            −      4              m        2                    (      4              m        2            −      2      m      +      1      )      (      2      m      −      1              )        2            Factoring one last time:            4              m        2            (      m      −      1      )              (      4              m        2            −      2      m      +      1      )      (      2      m      −      1              )        2