\( \left. \begin{array} { l l l } { + 24 x ^ { 4 } : ( - 12 x ^ { 3 } ) } & { + 48 x ^ { 5 } y ^ { 2 } z : ( + 2 y ^ { 2 } z ) } & { - 18 x ^ { 3 } y ^ { 2 } z : ( - 9 x ^ { 3 } y ^ { 2 } z ) } \\ { - 36 a ^ { 5 } : ( - 18 a ) } & { - 27 a ^ { 2 } b c ^ { 2 } : ( + 9 a ^ { 2 } b ) } & { + 42 a ^ { 9 } x y ^ { 2 } : ( + 21 a ^ { 9 } y ^ { 2 } ) } \end{array} \right. \)

Let's break down each expression step by step. 1. **First Expression**: \( \frac{+ 24 x^4}{- 12 x^3} \) - Dividing the coefficients: \( \frac{24}{-12} = -2 \) - For the variable \( x \): \( x^4 \div x^3 = x^{4-3} = x^1 = x \) - Result: \( -2x \) 2. **Second Expression**: \( \frac{+ 48 x^5 y^2 z}{+ 2 y^2 z} \) - Dividing the coefficients: \( \frac{48}{2} = 24 \) - For \( x \): \( x^5 \) remains since there's no \( x \) in the denominator. - For \( y^2 \): \( y^2 \div y^2 = 1 \) - For \( z \): \( z \div z = 1 \) - Result: \( 24 x^5 \) 3. **Third Expression**: \( \frac{- 18 x^3 y^2 z}{- 9 x^3 y^2 z} \) - Dividing the coefficients: \( \frac{-18}{-9} = 2 \) - For \( x^3 \): \( x^3 \div x^3 = 1 \) - For \( y^2 \): \( y^2 \div y^2 = 1 \) - For \( z \): \( z \div z = 1 \) - Result: \( 2 \) 4. **Fourth Expression**: \( \frac{- 36 a^5}{- 18 a} \) - Dividing the coefficients: \( \frac{-36}{-18} = 2 \) - For \( a^5 \): \( a^5 \div a^1 = a^{5-1} = a^4 \) - Result: \( 2a^4 \) 5. **Fifth Expression**: \( \frac{- 27 a^2 b c^2}{+ 9 a^2 b} \) - Dividing the coefficients: \( \frac{-27}{9} = -3 \) - For \( a^2 \): \( a^2 \div a^2 = 1 \) - For \( b \): \( b \div b = 1 \) - \( c^2 \) remains since there's no \( c \) in the denominator. - Result: \( -3c^2 \) 6. **Sixth Expression**: \( \frac{+ 42 a^9 x y^2}{+ 21 a^9 y^2} \) - Dividing the coefficients: \( \frac{42}{21} = 2 \) - For \( a^9 \): \( a^9 \div a^9 = 1 \) - For \( y^2 \): \( y^2 \div y^2 = 1 \) - \( x \) remains since there's no \( x \) in the denominator. - Result: \( 2x \) Putting it all together, the results are: - 1) \( -2x \) - 2) \( 24x^5 \) - 3) \( 2 \) - 4) \( 2a^4 \) - 5) \( -3c^2 \) - 6) \( 2x \)