Cubic Equation and Inequality

Algebra Level 5

\[ \large \frac{\sqrt{(\lambda +3)^2 + (\rho +3)^2 +18} + 3\lambda +3\rho +1}{\sqrt{\lambda^2 + \rho^2} +1} \geq 7\]

Let \(\lambda, \epsilon, \rho\) be the roots of the equation \((x+a)^3 = 0\) for real number \(a\). If the inequality above is fulfilled, find the numeric value of the expression below.

\[ \large \frac{\sqrt{(\lambda +3)^2 + (\rho +3)^2 +18} + 3\lambda +3\rho +1}{\sqrt{\lambda^2 + \rho^2} +1} \]

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