To find \((f+g+h)(x)\), we need to add together the functions \(f(x)\), \(g(x)\), and \(h(x)\).
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The function \(f(x)\) is given as: \[ f(x) = -2|x + 2| - 1 \]
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The function \(g(x)\) is given as: \[ g(x) = x^2 + 3 \]
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The function \(h(x)\) is given as: \[ h(x) = \sqrt{4x - 2} + 3 \]
Now, we add these functions together:
\[ (f + g + h)(x) = f(x) + g(x) + h(x) \]
Substituting each function into the equation:
\[ (f + g + h)(x) = (-2|x + 2| - 1) + (x^2 + 3) + (\sqrt{4x - 2} + 3) \]
Combining like terms:
\[ = -2|x + 2| + x^2 + \sqrt{4x - 2} + 3 - 1 + 3 \] \[ = -2|x + 2| + x^2 + \sqrt{4x - 2} + 5 \]
Thus, the final expression for \((f+g+h)(x)\) is:
\[ \boxed{-2|x + 2| + x^2 + \sqrt{4x - 2} + 5} \]