To solve this equation, we first need to simplify it by combining like terms:

4x² - 6x - 1 + x√(2x² + 6x + 1) = 0

Next, we can isolate the square root term on one side of the equation:

x√(2x² + 6x + 1) = -4x² + 6x + 1

Now, square both sides of the equation to eliminate the square root:

x²(2x² + 6x + 1) = (-4x² + 6x + 1)²

Expand both sides:

2x^4 + 6x^3 + x² = 16x^4 - 48x³ + 12x² - 24x² + 36x + 1

Combine like terms:

2x^4 + 6x^3 + x² = 16x^4 - 48x³ - 12x² + 36x + 1

Rearrange the equation to set it equal to zero:

14x^4 - 54x³ - 13x² + 36x + 1 = 0

This is a quartic equation that can be solved using various methods such as factoring, graphing, or numerical methods.