1. **State the problem:** Simplify the expression $$\left(2x - 4x^2 - 12x + 1\right) \cdot 2x + 4x^2 - 12x + 1 \cdot 2x + 4x^2 - 12x + 1.$$\n\n2. **Rewrite the expression clearly:** The expression can be interpreted as $$\left(2x - 4x^2 - 12x + 1\right) \cdot 2x + \left(4x^2 - 12x + 1\right) \cdot \left(2x + 4x^2 - 12x + 1\right).$$\n\n3. **Simplify the first part:** Combine like terms inside the first parentheses: $$2x - 4x^2 - 12x + 1 = -4x^2 - 10x + 1.$$\nMultiply by $2x$: $$(-4x^2 - 10x + 1) \cdot 2x = -8x^3 - 20x^2 + 2x.$$\n\n4. **Simplify the second part:** Multiply $$\left(4x^2 - 12x + 1\right) \cdot \left(2x + 4x^2 - 12x + 1\right).$$\n\n5. **Multiply the polynomials:**\n- $4x^2 \cdot 2x = 8x^3$\n- $4x^2 \cdot 4x^2 = 16x^4$\n- $4x^2 \cdot (-12x) = -48x^3$\n- $4x^2 \cdot 1 = 4x^2$\n- $-12x \cdot 2x = -24x^2$\n- $-12x \cdot 4x^2 = -48x^3$\n- $-12x \cdot (-12x) = 144x^2$\n- $-12x \cdot 1 = -12x$\n- $1 \cdot 2x = 2x$\n- $1 \cdot 4x^2 = 4x^2$\n- $1 \cdot (-12x) = -12x$\n- $1 \cdot 1 = 1$\n\n6. **Combine like terms:**\n- $16x^4$\n- $8x^3 - 48x^3 - 48x^3 = 8x^3 - 96x^3 = -88x^3$\n- $4x^2 - 24x^2 + 144x^2 + 4x^2 = (4 - 24 + 144 + 4)x^2 = 128x^2$\n- $-12x + 2x - 12x = (-12 + 2 - 12)x = -22x$\n- Constant term: $1$\n\n7. **Sum the two parts:**\n$$(-8x^3 - 20x^2 + 2x) + (16x^4 - 88x^3 + 128x^2 - 22x + 1) = 16x^4 + (-8x^3 - 88x^3) + (-20x^2 + 128x^2) + (2x - 22x) + 1 = 16x^4 - 96x^3 + 108x^2 - 20x + 1.$$\n\n**Final answer:** $$16x^4 - 96x^3 + 108x^2 - 20x + 1.$$