Question

(4x - 2)^2 + (- 2x^3 + 8x - 4)

Explanation:

Step1: Expand the square term

We use the formula \((a - b)^2 = a^2 - 2ab + b^2\) where \(a = 4x\) and \(b=2\). So \((4x - 2)^2=(4x)^2-2\times4x\times2 + 2^2=16x^2-16x + 4\).

Step2: Combine with the polynomial

Now we add the expanded term to \(-2x^3 + 8x - 4\). So we have \(16x^2-16x + 4-2x^3 + 8x - 4\).

Step3: Simplify the polynomial

Combine like terms: the constant terms \(4\) and \(-4\) cancel out. The \(x\) terms: \(-16x+8x=-8x\). So the simplified form is \(-2x^3 + 16x^2-8x\). We can also factor out \(-2x\) to get \(-2x(x^2 - 8x + 4)\) (optional, but the simplified polynomial is \(-2x^3+16x^2 - 8x\)).

Answer:

\(-2x^3 + 16x^2-8x\) (or factored form \(-2x(x^2 - 8x + 4)\))