Find the largest integral value of ' \( a \) ' for which every solu...

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Find the largest integral value of ' \( a \) ' for which every solution of the equation \( x([x]-5)+2\{x\}+6=0 \) satisfies the inequality \( (a-3) x^{2}+2(a+3) x-8 a \leq 0 \), where
\( \mathrm{P} \) \( [y] \) and \( \{y\} \) denote greatest integer and fractional part functions respectively.

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