Cam someboy please help me out with this? I've done a part of it so far. This is the question:

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Given the polynomial $\displaystyle P(x) = x^4 + ax^3 + bx^2 + cx - 10$ has real coefficients.

IF we know $\displaystyle P(x)$ has a complex zero $\displaystyle 1 + ki$ (k being an integer), use this to find an expression for a real quadratic factor of $\displaystyle P(x)$.

Hence, find all the possible values for $\displaystyle k$ for which the quadratic is a factor of $\displaystyle P(x)$

Now, this is what I have done so far:

If $\displaystyle 1 + ki$ is a zero, then $\displaystyle 1 - ki$ is also a zero (conjugates).

From this, I used the sum and product rules:

SUM : $\displaystyle (1 + ki) + (1 - ki) = 2$

PRODUCT : $\displaystyle (1 + ki)(1 - ki) = 1 - k^2$

This gives $\displaystyle x^2 - 2x + (1 - k^2)$

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What do I do with this? Now that I have a quadratic factor, I'm supposed to find the possible values of k. I was thinking maybe $\displaystyle k > 0$ and that's the answer but I'm not sure (square rooting negative number gives an imaginary solution)

Can someone check it for me?

Thanks. All help is appreciated.