$$(z^{4}+6z^{3}+10z^{2}+6z+1)+(-z^{4}+6z^{3}-10z^{2}+6z-1)$$
$12z\left(z^{2}+1\right)$
$$z^{4}+12z^{3}+10z^{2}+6z+1-z^{4}-10z^{2}+6z-1$$
$$z^{4}+12z^{3}+6z+1-z^{4}+6z-1$$
$$z^{4}+12z^{3}+12z+1-z^{4}-1$$
$$z^{4}+12z^{3}+12z-z^{4}$$
$$12z^{3}+12z$$
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$$12\left(z^{3}+z\right)$$
$$z\left(z^{2}+1\right)$$
$$12z\left(z^{2}+1\right)$$