Solve for x over the real numbers:

abs(2 x^2 + 3 x - 1)^(3 x^2 - 13 x + 4) = 1

Take the natural logarithm of both sides:

log(abs(2 x^2 + 3 x - 1)) (3 x^2 - 13 x + 4) = 0

The left hand side factors into a product with three terms:

log(abs(2 x^2 + 3 x - 1)) (x - 4) (3 x - 1) = 0

Split (x - 4) (3 x - 1) log(abs(2 x^2 + 3 x - 1)) into separate parts with additional assumptions.

Assume abs(2 x^2 + 3 x - 1)!=0 from log(abs(2 x^2 + 3 x - 1)):

x - 4 = 0 for abs(2 x^2 + 3 x - 1)!=0

or 3 x - 1 = 0 for abs(2 x^2 + 3 x - 1)!=0

or log(abs(2 x^2 + 3 x - 1)) = 0

Add 4 to both sides:

x = 4 or 3 x - 1 = 0 for abs(2 x^2 + 3 x - 1)!=0

or log(abs(2 x^2 + 3 x - 1)) = 0

Add 1 to both sides:

x = 4 or 3 x = 1 for abs(2 x^2 + 3 x - 1)!=0

or log(abs(2 x^2 + 3 x - 1)) = 0

Divide both sides by 3:

x = 4 or x = 1/3 or log(abs(2 x^2 + 3 x - 1)) = 0

Cancel logarithms by taking exp of both sides:

x = 4 or x = 1/3 or abs(2 x^2 + 3 x - 1) = 1

Split the equation into two possible cases:

x = 4 or x = 1/3 or 2 x^2 + 3 x - 1 = 1 or 2 x^2 + 3 x - 1 = -1

Divide both sides by 2:

x = 4 or x = 1/3 or x^2 + (3 x)/2 - 1/2 = 1/2 or 2 x^2 + 3 x - 1 = -1

Add 1/2 to both sides:

x = 4 or x = 1/3 or x^2 + (3 x)/2 = 1 or 2 x^2 + 3 x - 1 = -1

Add 9/16 to both sides:

x = 4 or x = 1/3 or x^2 + (3 x)/2 + 9/16 = 25/16 or 2 x^2 + 3 x - 1 = -1

Write the left hand side as a square:

x = 4 or x = 1/3 or (x + 3/4)^2 = 25/16 or 2 x^2 + 3 x - 1 = -1

Take the square root of both sides:

x = 4 or x = 1/3 or x + 3/4 = 5/4 or x + 3/4 = -5/4 or 2 x^2 + 3 x - 1 = -1

Subtract 3/4 from both sides:

x = 4 or x = 1/3 or x = 1/2 or x + 3/4 = -5/4 or 2 x^2 + 3 x - 1 = -1

Subtract 3/4 from both sides:

x = 4 or x = 1/3 or x = 1/2 or x = -2 or 2 x^2 + 3 x - 1 = -1

Divide both sides by 2:

x = 4 or x = 1/3 or x = 1/2 or x = -2 or x^2 + (3 x)/2 - 1/2 = -1/2

Add 1/2 to both sides:

x = 4 or x = 1/3 or x = 1/2 or x = -2 or x^2 + (3 x)/2 = 0

Factor x and constant terms from the left hand side:

x = 4 or x = 1/3 or x = 1/2 or x = -2 or 1/2 x (2 x + 3) = 0

Multiply both sides by 2:

x = 4 or x = 1/3 or x = 1/2 or x = -2 or x (2 x + 3) = 0

Split into two equations:

x = 4 or x = 1/3 or x = 1/2 or x = -2 or x = 0 or 2 x + 3 = 0

Subtract 3 from both sides:

x = 4 or x = 1/3 or x = 1/2 or x = -2 or x = 0 or 2 x = -3

Divide both sides by 2:

**x = 4 or x = 1/3 or x = 1/2 or x = -2 or x = 0 or x = -3/2**