1) \( F_{x}=11^{3}+x^{6}+8 x^{5}+3 x^{4}+2 x^{3}+2 x^{2}-5 \) \( g_{x}=2 x^{5}-4 x^{4}-3 x^{3}+2 x^{2}+4 x-7 \) (i) \( F_{x}+g x \) (ii) \( F_{x-g x} \) (ii) \( g_{x}-F_{x} \)

To solve the expressions involving \( F_{x} \) and \( g_{x} \), we first calculate \( F_{x} + g_{x} \). 1. \( F_{x} + g_{x} \): \[ F_{x} + g_{x} = \left( 11^{3} + x^{6} + 8x^{5} + 3x^{4} + 2x^{3} + 2x^{2} - 5 \right) + \left( 2x^{5} - 4x^{4} - 3x^{3} + 2x^{2} + 4x - 7 \right) \] Combining like terms, we get: \[ F_{x} + g_{x} = 11^{3} + x^{6} + (8x^{5} + 2x^{5}) + (3x^{4} - 4x^{4}) + (2x^{3} - 3x^{3}) + (2x^{2} + 2x^{2}) + 4x - 5 - 7 \] \[ = 11^{3} + x^{6} + 10x^{5} - x^{4} - x^{3} + 4x^{2} + 4x - 12 \] 2. \( F_{x-g_{x}} \): We need to express this as \( F_{(x - (2x^{5} - 4x^{4} - 3x^{3} + 2x^{2} + 4x - 7))} \): \[ x - g_{x} = x - (2x^{5} - 4x^{4} - 3x^{3} + 2x^{2} + 4x - 7) = -2x^{5} + 4x^{4} + 3x^{3} - x^{2} - 3x + 7 \] Now, substitute this expression into \( F_{x} \): \[ F_{x - g_{x}} = 11^{3} + (-2x^{5} + 4x^{4} + 3x^{3} - x^{2} - 3x + 7)^{6} + 8(-2x^{5} + 4x^{4} + 3x^{3} - x^{2} - 3x + 7)^{5} + ... \] (The calculation will involve expanding the power and is quite lengthy.) 3. \( g_{x} - F_{x} \): \[ g_{x} - F_{x} = \left( 2x^{5} - 4x^{4} - 3x^{3} + 2x^{2} + 4x - 7 \right) - \left( 11^{3} + x^{6} + 8x^{5} + 3x^{4} + 2x^{3} + 2x^{2} - 5 \right) \] \[ = -x^{6} + (2x^{5} - 8x^{5}) + (-4x^{4} - 3x^{4}) + (-3x^{3} - 2x^{3}) + (2x^{2} - 2x^{2}) + (4x - 0) - (11^{3} - (-5) - 7) \] \[ = -x^{6} - 6x^{5} - 7x^{4} - 5x^{3} + 4x - (11^{3} - 2) \] That sums up the evaluations! Remember to provide the algebraic simplifications for each case as needed!