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

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

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

Reorder the terms:
(1 + 2a)(a + 3) + 3 = 0

Reorder the terms:
(1 + 2a)(3 + a) + 3 = 0

Multiply (1 + 2a) * (3 + a)
(1(3 + a) + 2a * (3 + a)) + 3 = 0
((3 * 1 + a * 1) + 2a * (3 + a)) + 3 = 0
((3 + 1a) + 2a * (3 + a)) + 3 = 0
(3 + 1a + (3 * 2a + a * 2a)) + 3 = 0
(3 + 1a + (6a + 2a2)) + 3 = 0

Combine like terms: 1a + 6a = 7a
(3 + 7a + 2a2) + 3 = 0

Reorder the terms:
3 + 3 + 7a + 2a2 = 0

Combine like terms: 3 + 3 = 6
6 + 7a + 2a2 = 0

Solving
6 + 7a + 2a2 = 0

Solving for variable 'a'.

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

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

Simplifying
2 + a = 0

Solving
2 + a = 0

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

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

Combine like terms: 2 + -2 = 0
0 + a = 0 + -2
a = 0 + -2

Combine like terms: 0 + -2 = -2
a = -2

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

Simplifying
3 + 2a = 0

Solving
3 + 2a = 0

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

Add '-3' to each side of the equation.
3 + -3 + 2a = 0 + -3

Combine like terms: 3 + -3 = 0
0 + 2a = 0 + -3
2a = 0 + -3

Combine like terms: 0 + -3 = -3
2a = -3

Divide each side by '2'.
a = -1.5

Simplifying
a = -1.5
Solution
a = {-2, -1.5}