The characteristic polynomial is given by

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

In this example we have:

$$ \begin{aligned} p(x) &= det(A - x I) = det \left( \left[ \begin{matrix}-3&0&4\\0&-1&0\\2&7&3\end{matrix} \right] - \left[ \begin{matrix} \lambda & 0 & 0 \\ 0 & \lambda & 0 \\ 0 & 0 & \lambda \end{matrix} \right] \right) = \\ &= \left| \begin{matrix}-3 - \lambda &0&4\\0&-1 - \lambda &0\\2&7&3 - \lambda \end{matrix} \right| = \\ &= (-\lambda-3) (-\lambda-1) (-\lambda+3) + 0 \cdot 0 \cdot 2 + 4 \cdot 0 \cdot 7 - (-\lambda-3) \cdot 0 \cdot 7 - 0 \cdot 0 \cdot (-\lambda+3) - 2 \cdot (-\lambda-1) \cdot 4 = \\ & = -\lambda^3-\lambda^2+9\lambda+9 + 0 + 0 - 0 - 0 - (-8\lambda-8) = \\ & = -\lambda^3-\lambda^2+17\lambda+17 \end{aligned} $$

Characteristic polynomial calculator