The complex zeros of the polynomial \(f(x)\) are \(x = 1\) and \(x = -8\). Both of these values are real numbers; there are no complex zeros for this polynomial.
To find the complex zeros of the given polynomial \(f(x) = x + x^2 + 2x^2 + 4x - 8\), let's first combine like terms and set the expression equal to zero:
\[ f(x) = x^2 + 5x - 8 \]
Now, we want to find the values of \(x\) that make \(f(x)\) equal to zero. To do this, we can factor the quadratic:
\[ (x - 1)(x + 8) = 0 \]
Now, we have two factors, and we can set each factor equal to zero:
\[ x - 1 = 0 \implies x = 1 \]
\[ x + 8 = 0 \implies x = -8 \]
So, the complex zeros of the polynomial \(f(x)\) are \(x = 1\) and \(x = -8\). Both of these values are real numbers; there are no complex zeros for this polynomial.