(a+b)5a+(a+b)3b=0

Solution for (a+b)5a+(a+b)3b=0 equation:

Simplifying
(a + b) * 5a + (a + b) * 3b = 0

Reorder the terms for easier multiplication:
5a(a + b) + (a + b) * 3b = 0
(a * 5a + b * 5a) + (a + b) * 3b = 0

Reorder the terms:
(5ab + 5a2) + (a + b) * 3b = 0
(5ab + 5a2) + (a + b) * 3b = 0

Reorder the terms for easier multiplication:
5ab + 5a2 + 3b(a + b) = 0
5ab + 5a2 + (a * 3b + b * 3b) = 0
5ab + 5a2 + (3ab + 3b2) = 0

Reorder the terms:
5ab + 3ab + 5a2 + 3b2 = 0

Combine like terms: 5ab + 3ab = 8ab
8ab + 5a2 + 3b2 = 0

Solving
8ab + 5a2 + 3b2 = 0

Solving for variable 'a'.

Factor a trinomial.
(5a + 3b)(a + b) = 0

Subproblem 1
Set the factor '(5a + 3b)' equal to zero and attempt to solve:

Simplifying
5a + 3b = 0

Solving
5a + 3b = 0

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

Add '-3b' to each side of the equation.
5a + 3b + -3b = 0 + -3b

Combine like terms: 3b + -3b = 0
5a + 0 = 0 + -3b
5a = 0 + -3b
Remove the zero:
5a = -3b

Divide each side by '5'.
a = -0.6b

Simplifying
a = -0.6b
Subproblem 2
Set the factor '(a + b)' equal to zero and attempt to solve:

Simplifying
a + b = 0

Solving
a + b = 0

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

Add '-1b' to each side of the equation.
a + b + -1b = 0 + -1b

Combine like terms: b + -1b = 0
a + 0 = 0 + -1b
a = 0 + -1b
Remove the zero:
a = -1b

Simplifying
a = -1b
Solution
a = {-0.6b, -1b}