19=[(a+1)*(a+1)*(a+1)]-[a*a*a]

Solution for 19=[(a+1)*(a+1)*(a+1)]-[a*a*a] equation:

Simplifying
19 = [(a + 1)(a + 1)(a + 1)] + -1[a * a * a]

Reorder the terms:
19 = [(1 + a)(a + 1)(a + 1)] + -1[a * a * a]

Reorder the terms:
19 = [(1 + a)(1 + a)(a + 1)] + -1[a * a * a]

Reorder the terms:
19 = [(1 + a)(1 + a)(1 + a)] + -1[a * a * a]

Multiply (1 + a) * (1 + a)
19 = [(1(1 + a) + a(1 + a))(1 + a)] + -1[a * a * a]
19 = [((1 * 1 + a * 1) + a(1 + a))(1 + a)] + -1[a * a * a]
19 = [((1 + 1a) + a(1 + a))(1 + a)] + -1[a * a * a]
19 = [(1 + 1a + (1 * a + a * a))(1 + a)] + -1[a * a * a]
19 = [(1 + 1a + (1a + a2))(1 + a)] + -1[a * a * a]

Combine like terms: 1a + 1a = 2a
19 = [(1 + 2a + a2)(1 + a)] + -1[a * a * a]

Multiply (1 + 2a + a2) * (1 + a)
19 = [(1(1 + a) + 2a * (1 + a) + a2(1 + a))] + -1[a * a * a]
19 = [((1 * 1 + a * 1) + 2a * (1 + a) + a2(1 + a))] + -1[a * a * a]
19 = [((1 + 1a) + 2a * (1 + a) + a2(1 + a))] + -1[a * a * a]
19 = [(1 + 1a + (1 * 2a + a * 2a) + a2(1 + a))] + -1[a * a * a]
19 = [(1 + 1a + (2a + 2a2) + a2(1 + a))] + -1[a * a * a]
19 = [(1 + 1a + 2a + 2a2 + (1 * a2 + a * a2))] + -1[a * a * a]
19 = [(1 + 1a + 2a + 2a2 + (1a2 + a3))] + -1[a * a * a]

Combine like terms: 1a + 2a = 3a
19 = [(1 + 3a + 2a2 + 1a2 + a3)] + -1[a * a * a]

Combine like terms: 2a2 + 1a2 = 3a2
19 = [(1 + 3a + 3a2 + a3)] + -1[a * a * a]
19 = [1 + 3a + 3a2 + a3] + -1[a * a * a]

Remove brackets around [1 + 3a + 3a2 + a3]
19 = 1 + 3a + 3a2 + a3 + -1[a * a * a]

Multiply a * a
19 = 1 + 3a + 3a2 + a3 + -1[a2 * a]

Multiply a2 * a
19 = 1 + 3a + 3a2 + a3 + -1[a3]

Combine like terms: a3 + -1a3 = 0
19 = 1 + 3a + 3a2 + 0
19 = 1 + 3a + 3a2

Solving
19 = 1 + 3a + 3a2

Solving for variable 'a'.

Combine like terms: 19 + -1 = 18
18 + -3a + -3a2 = 1 + 3a + 3a2 + -1 + -3a + -3a2

Reorder the terms:
18 + -3a + -3a2 = 1 + -1 + 3a + -3a + 3a2 + -3a2

Combine like terms: 1 + -1 = 0
18 + -3a + -3a2 = 0 + 3a + -3a + 3a2 + -3a2
18 + -3a + -3a2 = 3a + -3a + 3a2 + -3a2

Combine like terms: 3a + -3a = 0
18 + -3a + -3a2 = 0 + 3a2 + -3a2
18 + -3a + -3a2 = 3a2 + -3a2

Combine like terms: 3a2 + -3a2 = 0
18 + -3a + -3a2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(6 + -1a + -1a2) = 0

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

Ignore the factor 3.
Subproblem 1
Set the factor '(2 + -1a)' equal to zero and attempt to solve:

Simplifying
2 + -1a = 0

Solving
2 + -1a = 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 + -1a = 0 + -2

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

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

Divide each side by '-1'.
a = 2

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

Simplifying
3 + a = 0

Solving
3 + a = 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 + a = 0 + -3

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

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

Simplifying
a = -3
Solution
a = {2, -3}