The fully factored form is $ oxed{(3x - 5y - 4z)(3x - 5y + 4z)} $.Question: Compute the sum of the roots of the equation $ u^{3/2} - 4u + 3\sqrt{u} = 0 $, given that all roots are non-negative real numbers.

["The Fully Factored Form and the Sum of Roots: Solving the Equation $ u^{3/2} - 4u + 3\sqrt{u} = 0 $", "Understanding polynomial and radical equations begins with recognizing and simplifying complex forms — and sometimes, factoring isn’t just about polynomials. Today, we explore the fully factored expression $ \boxed{(3x - 5y - 4z)(3x - 5y + 4z)} $, before diving deep into solving a powerful equation involving square roots, computing the sum of its non-negative real roots.", "---", "### Factoring the Expression", "We are given the fully factored form:\n[\n(3x - 5y - 4z)(3x - 5y + 4z)\n]\nThis resembles the identity $ (a - b)(a + b) = a^2 - b^2 $. However, here the first bracket resembles $ a - b - c $, but carefully observing, if we let $ a = 3x - 5y $ and $ b = 4z $, the expression becomes:\n[\n(a - 4z)(a + 4z) = a^2 - (4z)^2 = (3x - 5y)^2 - (4z)^2\n]\nThus, the expression fully expands to:\n[\n(3x - 5y)^2 - 16z^2\n]\nBut more importantly, this factorization confirms that the original equation may be rewritten in terms of variables related to square roots — a theme resonating with radical equations.", "---", "### Solving the Radical Equation: $ u^{3/2} - 4u + 3\sqrt{u} = 0 $", "Let us analyze the equation:\n[\nu^{3/2} - 4u + 3\sqrt{u} = 0\n]\nSince the equation involves $ u^{3/2} = u \cdot \sqrt{u} $ and $ \sqrt{u} $, define a substitution:\nLet $ x = \sqrt{u} $. Then $ u = x^2 $ and $ u^{3/2} = x^3 $. Substituting:\n[\nx^3 - 4x^2 + 3x = 0\n]\nFactor out $ x $:\n[\nx(x^2 - 4x + 3) = 0\n]\nNow factor the quadratic:\n[\nx^2 - 4x + 3 = (x - 1)(x - 3)\n]\nSo the equation becomes:\n[\nx(x - 1)(x - 3) = 0\n]\nSolutions are $ x = 0 $, $ x = 1 $, and $ x = 3 $.", "Since $ x = \sqrt{u} $, and $ u \geq 0 $, all solutions are valid as long as $ x \geq 0 $ — which they are. Convert back to $ u $:\n- $ x = 0 \Rightarrow u = 0^2 = 0 $\n- $ x = 1 \Rightarrow u = 1^2 = 1 $\n- $ x = 3 \Rightarrow u = 3^2 = 9 $", "Thus, the non-negative real roots are $ u = 0, 1, 9 $.", "---", "### Sum of the Roots", "The sum is:\n[\n0 + 1 + 9 = 10\n]", "This elegant result — arriving at the sum of roots via factoring and substitution — demonstrates how substitution transforms complex radical equations into solvable algebraic forms. Notably, while the fully factored expression $ (3x - 5y - 4z)(3x - 5y + 4z) $ was introduced in a broader context, the core solving technique here reflects substitution methods invaluable in Olympiad and advanced algebra.", "---", "### Final Answer", "The sum of the non-negative real roots of the equation $ u^{3/2} - 4u + 3\sqrt{u} = 0 $ is\n[\n\boxed{10}\n]"]