v(2v+1)+3(2v+1)=0

Solution for v(2v+1)+3(2v+1)=0 equation:

Simplifying
v(2v + 1) + 3(2v + 1) = 0

Reorder the terms:
v(1 + 2v) + 3(2v + 1) = 0
(1 * v + 2v * v) + 3(2v + 1) = 0
(1v + 2v2) + 3(2v + 1) = 0

Reorder the terms:
1v + 2v2 + 3(1 + 2v) = 0
1v + 2v2 + (1 * 3 + 2v * 3) = 0
1v + 2v2 + (3 + 6v) = 0

Reorder the terms:
3 + 1v + 6v + 2v2 = 0

Combine like terms: 1v + 6v = 7v
3 + 7v + 2v2 = 0

Solving
3 + 7v + 2v2 = 0

Solving for variable 'v'.

Factor a trinomial.
(3 + v)(1 + 2v) = 0

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

Simplifying
3 + v = 0

Solving
3 + v = 0

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

Add '-3' to each side of the equation.
3 + -3 + v = 0 + -3

Combine like terms: 3 + -3 = 0
0 + v = 0 + -3
v = 0 + -3

Combine like terms: 0 + -3 = -3
v = -3

Simplifying
v = -3
Subproblem 2
Set the factor '(1 + 2v)' equal to zero and attempt to solve:

Simplifying
1 + 2v = 0

Solving
1 + 2v = 0

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

Add '-1' to each side of the equation.
1 + -1 + 2v = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 2v = 0 + -1
2v = 0 + -1

Combine like terms: 0 + -1 = -1
2v = -1

Divide each side by '2'.
v = -0.5

Simplifying
v = -0.5
Solution
v = {-3, -0.5}