**Answer:**

**Step-by-step explanation:**

(3m+2n)/(m-n2)+(5m+n)/(n-m2)-(2m-3n)/(m2-n)

Final result :

-3m3 - 2m2n + 7m2 - 7mn2 + mn + 2n3 + 2n2

—————————————————————————————————————————

(m - n2) • (n - m2)

Step by step solution :

Step 1 :

2m - 3n

Simplify ———————

m2 - n

Trying to factor as a Difference of Squares :

1.1 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

Equation at the end of step 1 :

(3m+2n) (5m+n) (2m-3n)

(————————+————————)-———————

(m-(n2)) (n-(m2)) m2-n

Step 2 :

5m + n

Simplify ——————

n - m2

Trying to factor as a Difference of Squares :

2.1 Factoring: n - m2

Check : n1 is not a square !!

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

Equation at the end of step 2 :

(3m+2n) (5m+n) (2m-3n)

(————————+——————)-———————

(m-(n2)) n-m2 m2-n

Step 3 :

3m + 2n

Simplify ———————

m - n2

Trying to factor as a Difference of Squares :

3.1 Factoring: m - n2

Check : m1 is not a square !!

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

Equation at the end of step 3 :

(3m + 2n) (5m + n) (2m - 3n)

(————————— + ————————) - —————————

m - n2 n - m2 m2 - n

Step 4 :

Calculating the Least Common Multiple :

4.1 Find the Least Common Multiple

The left denominator is : m-n2

The right denominator is : n-m2

Number of times each Algebraic Factor

appears in the factorization of:

Algebraic

Factor Left

Denominator Right

Denominator L.C.M = Max

{Left,Right}

m-n2 1 0 1

n-m2 0 1 1

Least Common Multiple:

(m-n2) • (n-m2)

Calculating Multipliers :

4.2 Calculate multipliers for the two fractions

Denote the Least Common Multiple by L.C.M

Denote the Left Multiplier by Left_M

Denote the Right Multiplier by Right_M

Denote the Left Deniminator by L_Deno

Denote the Right Multiplier by R_Deno

Left_M = L.C.M / L_Deno = n-m2

Right_M = L.C.M / R_Deno = m-n2