If \( 4 x^{3}-x=1 \), then what is \( 8 x^{4}+12 x^{3}-2 x^{2}-5 x+5 ? \)

To solve for \( 8x^4 + 12x^3 - 2x^2 - 5x + 5 \) given the equation \( 4x^3 - x = 1 \), we start by rearranging the equation: \[ 4x^3 = x + 1 \] Next, we can express \( x^3 \) in terms of \( x \): \[ x^3 = \frac{x + 1}{4} \] Now, we need to find \( x^4 \). We can do this by multiplying \( x^3 \) by \( x \): \[ x^4 = x \cdot x^3 = x \cdot \frac{x + 1}{4} = \frac{x^2 + x}{4} \] Now we can substitute \( x^3 \) and \( x^4 \) into the expression \( 8x^4 + 12x^3 - 2x^2 - 5x + 5 \): 1. Substitute \( x^4 \): \[ 8x^4 = 8 \cdot \frac{x^2 + x}{4} = 2(x^2 + x) = 2x^2 + 2x \] 2. Substitute \( x^3 \): \[ 12x^3 = 12 \cdot \frac{x + 1}{4} = 3(x + 1) = 3x + 3 \] Now we can combine these results into the expression: \[ 8x^4 + 12x^3 - 2x^2 - 5x + 5 = (2x^2 + 2x) + (3x + 3) - 2x^2 - 5x + 5 \] Now, combine like terms: \[ = 2x^2 + 2x + 3x + 3 - 2x^2 - 5x + 5 \] Combine the \( x^2 \) terms: \[ 2x^2 - 2x^2 = 0 \] Combine the \( x \) terms: \[ 2x + 3x - 5x = 0 \] Combine the constant terms: \[ 3 + 5 = 8 \] Thus, the entire expression simplifies to: \[ 8 \] Therefore, the value of \( 8x^4 + 12x^3 - 2x^2 - 5x + 5 \) is \[ \boxed{8} \]