(m+n/m-n)+(m-n/m+n) Final result : 2m2 ——— m

Step by step solution :Step  1  : n Simplify — m Equation at the end of step  1  : n n ((m+—)-n)+((m-—)+n) m m Step  2  :Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole 

Rewrite the whole as a fraction using  m  as the denominator :

m m • m m = — = ————— 1 m

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole 

Common denominator : The equivalent fraction and the other fraction INVOLVED in the calculation share the same denominator

Adding fractions that have a common denominator :

 2.2       Adding up the two equivalent fractions 

Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then REDUCE to lowest terms if possible:

m • m - (n) m2 - n ——————————— = —————— m m Equation at the end of step  2  : n (m2 - n) ((m + —) - n) + (———————— + n) m m Step  3  :Rewriting the whole as an Equivalent Fraction :

 3.1   Adding a whole to a fraction 

Rewrite the whole as a fraction using  m  as the denominator :

n n • m n = — = ————— 1 m Trying to factor as a Difference of Squares :

 3.2      Factoring:  m2 - n 

Theory : A difference of two perfect squares,  A2 - B2  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =

         A2 - AB + BA - B2 =

         A2 - AB + AB - B2 = 

         A2 - B2

Note :  AB = BA is the commutative property of multiplication. 

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check :  m2  is the SQUARE of  m1 

Check :  n1   is not a square !! 

Ruling : Binomial can not be factored as the difference of two perfect squares

Adding fractions that have a common denominator :

 3.3       Adding up the two equivalent fractions 

(m2-n) + n • m m2 + mn - n —————————————— = ——————————— m m Equation at the end of step  3  : n (m2 + mn - n) ((m + —) - n) + ————————————— m m Step  4  : n Simplify — m Equation at the end of step  4  : n (m2 + mn - n) ((m + —) - n) + ————————————— m m Step  5  :Rewriting the whole as an Equivalent Fraction :

 5.1   Adding a fraction to a whole 

Rewrite the whole as a fraction using  m  as the denominator :

m m • m m = — = ————— 1 m Adding fractions that have a common denominator :

 5.2       Adding up the two equivalent fractions 

m • m + n m2 + n ————————— = —————— m m Equation at the end of step  5  : (m2 + n) (m2 + mn - n) (———————— - n) + ————————————— m m Step  6  :Rewriting the whole as an Equivalent Fraction :

 6.1   Subtracting a whole from a fraction 

Rewrite the whole as a fraction using  m  as the denominator :

n n • m n = — = ————— 1 m Adding fractions that have a common denominator :

 6.2       Adding up the two equivalent fractions 

(m2+n) - (n • m) m2 - mn + n ———————————————— = ——————————— m m Equation at the end of step  6  : (m2 - mn + n) (m2 + mn - n) ————————————— + ————————————— m m Step  7  :Trying to factor a MULTI variable polynomial :

 7.1    Factoring    m2 - mn + n 

Try to factor this multi-variable trinomial using TRIAL and error 

 Factorization fails

Trying to factor a multi variable polynomial :

 7.2    Factoring    m2 + mn - n 

Try to factor this multi-variable trinomial using trial and error 

 Factorization fails

Adding fractions which have a common denominator :

 7.3       Adding fractions which have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

(m2-mn+n) + (m2+mn-n) 2m2 ————————————————————— = ——— m m Final result : 2m2 ——— m

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