(a)(a+1)=(a+(a+1)+19)

Solution for (a)(a+1)=(a+(a+1)+19) equation:

Simplifying
(a)(a + 1) = (a + (a + 1) + 19)

Reorder the terms:
a(1 + a) = (a + (a + 1) + 19)
(1 * a + a * a) = (a + (a + 1) + 19)
(1a + a2) = (a + (a + 1) + 19)

Reorder the terms:
1a + a2 = (a + (1 + a) + 19)

Remove parenthesis around (1 + a)
1a + a2 = (a + 1 + a + 19)

Reorder the terms:
1a + a2 = (1 + 19 + a + a)

Combine like terms: 1 + 19 = 20
1a + a2 = (20 + a + a)

Combine like terms: a + a = 2a
1a + a2 = (20 + 2a)

Remove parenthesis around (20 + 2a)
1a + a2 = 20 + 2a

Solving
1a + a2 = 20 + 2a

Solving for variable 'a'.

Reorder the terms:
-20 + 1a + -2a + a2 = 20 + 2a + -20 + -2a

Combine like terms: 1a + -2a = -1a
-20 + -1a + a2 = 20 + 2a + -20 + -2a

Reorder the terms:
-20 + -1a + a2 = 20 + -20 + 2a + -2a

Combine like terms: 20 + -20 = 0
-20 + -1a + a2 = 0 + 2a + -2a
-20 + -1a + a2 = 2a + -2a

Combine like terms: 2a + -2a = 0
-20 + -1a + a2 = 0

Factor a trinomial.
(-4 + -1a)(5 + -1a) = 0

Subproblem 1
Set the factor '(-4 + -1a)' equal to zero and attempt to solve:

Simplifying
-4 + -1a = 0

Solving
-4 + -1a = 0

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

Add '4' to each side of the equation.
-4 + 4 + -1a = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -1a = 0 + 4
-1a = 0 + 4

Combine like terms: 0 + 4 = 4
-1a = 4

Divide each side by '-1'.
a = -4

Simplifying
a = -4
Subproblem 2
Set the factor '(5 + -1a)' equal to zero and attempt to solve:

Simplifying
5 + -1a = 0

Solving
5 + -1a = 0

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

Add '-5' to each side of the equation.
5 + -5 + -1a = 0 + -5

Combine like terms: 5 + -5 = 0
0 + -1a = 0 + -5
-1a = 0 + -5

Combine like terms: 0 + -5 = -5
-1a = -5

Divide each side by '-1'.
a = 5

Simplifying
a = 5
Solution
a = {-4, 5}