(m+2n)(5n+m)=

Solution for (m+2n)(5n+m)= equation:

Simplifying
(m + 2n)(5n + m) = 0

Reorder the terms:
(m + 2n)(m + 5n) = 0

Multiply (m + 2n) * (m + 5n)
(m(m + 5n) + 2n * (m + 5n)) = 0
((m * m + 5n * m) + 2n * (m + 5n)) = 0

Reorder the terms:
((5mn + m2) + 2n * (m + 5n)) = 0
((5mn + m2) + 2n * (m + 5n)) = 0
(5mn + m2 + (m * 2n + 5n * 2n)) = 0
(5mn + m2 + (2mn + 10n2)) = 0

Reorder the terms:
(5mn + 2mn + m2 + 10n2) = 0

Combine like terms: 5mn + 2mn = 7mn
(7mn + m2 + 10n2) = 0

Solving
7mn + m2 + 10n2 = 0

Solving for variable 'm'.

Factor a trinomial.
(m + 2n)(m + 5n) = 0

Subproblem 1
Set the factor '(m + 2n)' equal to zero and attempt to solve:

Simplifying
m + 2n = 0

Solving
m + 2n = 0

Move all terms containing m to the left, all other terms to the right.

Add '-2n' to each side of the equation.
m + 2n + -2n = 0 + -2n

Combine like terms: 2n + -2n = 0
m + 0 = 0 + -2n
m = 0 + -2n
Remove the zero:
m = -2n

Simplifying
m = -2n
Subproblem 2
Set the factor '(m + 5n)' equal to zero and attempt to solve:

Simplifying
m + 5n = 0

Solving
m + 5n = 0

Move all terms containing m to the left, all other terms to the right.

Add '-5n' to each side of the equation.
m + 5n + -5n = 0 + -5n

Combine like terms: 5n + -5n = 0
m + 0 = 0 + -5n
m = 0 + -5n
Remove the zero:
m = -5n

Simplifying
m = -5n
Solution
m = {-2n, -5n}