The characteristic polynomial is given by

$$ p(\lambda) = det(A - \lambda I) $$

In this example we have:

$$ \begin{aligned} p(x) &= det(A - x I) = det \left( \left[ \begin{matrix}4&2&2\\6&7&-2\\3&0&6\end{matrix} \right] - \left[ \begin{matrix} \lambda & 0 & 0 \\ 0 & \lambda & 0 \\ 0 & 0 & \lambda \end{matrix} \right] \right) = \\ &= \left| \begin{matrix}4 - \lambda &2&2\\6&7 - \lambda &-2\\3&0&6 - \lambda \end{matrix} \right| = \\ &= (-\lambda+4) (-\lambda+7) (-\lambda+6) + 2 \cdot (-2) \cdot 3 + 2 \cdot 6 \cdot 0 - (-\lambda+4) \cdot (-2) \cdot 0 - 6 \cdot 2 \cdot (-\lambda+6) - 3 \cdot (-\lambda+7) \cdot 2 = \\ & = -\lambda^3+17\lambda^2-94\lambda+168 + (-12) + 0 - 0 - (-12\lambda+72) - (-6\lambda+42) = \\ & = -\lambda^3+17\lambda^2-76\lambda+42 \end{aligned} $$

Characteristic polynomial calculator