\[ \begin{array}{l} =3\left(x^{2}+2 y^{2}\right)-\left[\begin{array}{l} 2 x^{2}+6 x^{3}+6 x y^{2} \\ = \\ =3 x\left(x^{2}+2 y^{2}\right) \end{array} \left\lvert\, \begin{array}{l} 33 a^{2} b-4 a b^{2}+112 b \\ -\frac{1}{2} a b\left(\frac{3}{2} a-2 b b+3\right) \end{array}\right.\right. \\ 411 a b c^{2}-22 a^{2} b+33 a b^{2} d \quad 5 \frac{3}{2} x^{2}+\frac{5}{2} x y-\frac{1}{2} y^{2} \\ =11 a b\left(c^{2}-2 a+3 b d\right)=\frac{1}{2}\left(3 x^{2}+5 x y-y^{2}\right) \\ \text { 6. } 12 a^{3} b^{2} c-16 a^{2} b^{2}+24 a^{4} b^{5} c^{3} \\ =4 a^{2} b^{2}\left(3 a c=4+6 a^{2} b^{3} c^{3}\right) \\ \text { 7. } 13 a^{4} b-26 b a^{3}+39 a^{2} b^{2} \quad \text { \& } 35 a+7 a b-21 a^{2} b+14 a b c \\ \left.=13 a^{2} b(2 a+3 b)^{-2 a+3 b}\right)=7 a(5+b-3 a b+2 b c \end{array} \] \( 94 a b c+16 a b-20 b c+24 a c \quad 10 \cdot 3 a(a+b)-4 b(a+b) \) \[ (a b c+4 a b-5 b c+6 a c)=(a+b)(3 a-4 b) \]

The problem presented appears to be a complex algebraic expression involving multiple variables and equations. Let's break it down step by step to clarify and simplify the expressions. ### Step 1: Identify the Components The expression contains several parts, including: 1. **Quadratic and cubic terms** in \(x\) and \(y\). 2. **Terms involving \(a\), \(b\), \(c\), and \(d\)**. 3. **Equations and equalities** that need to be simplified or solved. ### Step 2: Simplify Each Part Let's start by simplifying the first part of the expression: \[ 3\left(x^{2}+2 y^{2}\right) - \left(2 x^{2}+6 x^{3}+6 x y^{2}\right) \] This can be simplified as follows: 1. Distribute the \(3\): \[ 3x^{2} + 6y^{2} - (2x^{2} + 6x^{3} + 6xy^{2}) \] 2. Combine like terms: \[ (3x^{2} - 2x^{2}) + 6y^{2} - 6xy^{2} - 6x^{3} = x^{2} + 6y^{2} - 6xy^{2} - 6x^{3} \] ### Step 3: Analyze the Other Parts Next, we have the expression involving \(a\), \(b\), and \(c\): \[ 411abc^{2} - 22a^{2}b + 33ab^{2}d \] This expression does not seem to require simplification unless we have specific values for \(a\), \(b\), \(c\), and \(d\). ### Step 4: Solve the Equations Now, let's look at the equations provided: 1. \(11ab(c^{2} - 2a + 3bd) = \frac{1}{2}(3x^{2} + 5xy - y^{2})\) 2. \(12a^{3}b^{2}c - 16a^{2}b^{2} + 24a^{4}b^{5}c^{3}\) 3. \(4a^{2}b^{2}(3ac - 4 + 6a^{2}b^{3}c^{3})\) We can set up a system of equations based on these relationships. ### Step 5: Solve the System of Equations Let's solve the equations together. The equations can be represented as: 1. \(11ab(c^{2} - 2a + 3bd) - \frac{1}{2}(3x^{2} + 5xy - y^{2}) = 0\) 2. \(12a^{3}b^{2}c - 16a^{2}b^{2} + 24a^{4}b^{5}c^{3} = 0\) 3. \(4a^{2}b^{2}(3ac - 4 + 6a^{2}b^{3}c^{3}) = 0\) Now, I will use a tool to solve this system of equations. ### Step 6: Execute the Tool Call Let's proceed to solve the system of equations. ```json { "tool_uses": [ { "recipient_name": "functions.solve_system_equations", "parameters": { "equations": [ "11*a*b*(c^2 - 2*a + 3*b*d) - 1/2*(3*x^2 + 5*x*y - y^2) = 0", "12*a^3*b^2*c - 16*a^2*b^2 + 24*a^4*b^5*c^3 = 0", "4*a^2*b^2*(3*a*c - 4 + 6*a^2*b^3*c^3) = 0" ] } } ] } ``` Now, let's execute this to find the solutions for \(a\), \(b\), \(c\), \(d\), \(x\), and \(y\). ### Conclusion After solving the system of equations, we will have the values of the variables involved. If you have specific values for any of the variables or further instructions, please provide them for a more tailored solution.