c) \( \mathrm{mn}^{3} \) por \( \mathrm{m}^{3}-2 \mathrm{mn}-4 \mathrm{n}^{2} \) d) \( 2 \mathrm{x}^{2} \mathrm{y} \) por \( \mathrm{x}^{\mathrm{a}} \mathrm{y}^{\mathrm{b}}+\mathrm{x}^{\mathrm{a}-1} \mathrm{y}^{\mathrm{b}-1}-\mathrm{x}^{\mathrm{a}-2} \mathrm{y}^{\mathrm{b}+2} \) e) \( \frac{1}{2} m^{3} n^{4} \) por \( \frac{1}{3} m^{4}-m^{2} n^{2}+\frac{1}{4} n^{2} \) f) \( \frac{2}{5} y^{2} \) por \( \frac{1}{5} x^{2}-\frac{2}{3} x y+\frac{1}{4} y^{2} \) g) \( -2 \mathrm{x}^{2} \mathrm{y} \) por \( \mathrm{x}^{3}+5 \mathrm{x}^{2} \mathrm{y}^{2}-3 \mathrm{y}^{4} \) h) \( 3 \mathrm{~m}^{2} \) por \( \mathrm{m}^{\mathrm{a}+5}-4 \mathrm{~m}^{\mathrm{a}+3}-2 \mathrm{~m}^{\mathrm{a}+1}+5 \mathrm{~m}^{\mathrm{a}+4} \) i) \( -\frac{1}{4} x^{2} y^{2} z \) por \( 2 \mathrm{x}-3 \mathrm{yz}-5 \mathrm{z}^{2} \)

Simplify the expression by following steps: - step0: Multiply the terms: \(mn^{3}\left(m^{3}-2mn-4n^{2}\right)\) - step1: Apply the distributive property: \(mn^{3}m^{3}-mn^{3}\times 2mn-mn^{3}\times 4n^{2}\) - step2: Multiply the terms: \(m^{4}n^{3}-2m^{2}n^{4}-4mn^{5}\) Expand the expression \( -2x^{2}y(x^{3}+5x^{2}y^{2}-3y^{4}) \) Simplify the expression by following steps: - step0: Multiply the terms: \(-2x^{2}y\left(x^{3}+5x^{2}y^{2}-3y^{4}\right)\) - step1: Apply the distributive property: \(-2x^{2}yx^{3}-2x^{2}y\times 5x^{2}y^{2}-\left(-2x^{2}y\times 3y^{4}\right)\) - step2: Multiply the terms: \(-2x^{5}y-10x^{4}y^{3}-\left(-6x^{2}y^{5}\right)\) - step3: Remove the parentheses: \(-2x^{5}y-10x^{4}y^{3}+6x^{2}y^{5}\) Expand the expression \( \frac{1}{2} m^{3} n^{4}(\frac{1}{3} m^{4}-m^{2} n^{2}+\frac{1}{4} n^{2}) \) Simplify the expression by following steps: - step0: Simplify: \(\frac{1}{2}m^{3}n^{4}\left(\frac{1}{3}m^{4}-m^{2}n^{2}+\frac{1}{4}n^{2}\right)\) - step1: Rewrite the expression: \(2^{-1}\left(\frac{1}{3}m^{4}-m^{2}n^{2}+\frac{1}{4}n^{2}\right)m^{3}n^{4}\) - step2: Multiply the expression: \(\frac{1}{24}\left(4m^{4}-12m^{2}n^{2}+3n^{2}\right)m^{3}n^{4}\) - step3: Rearrange the terms: \(\frac{1}{6}m^{7}n^{4}-\frac{1}{2}m^{5}n^{6}+\frac{1}{8}n^{6}m^{3}\) Expand the expression \( \frac{2}{5} y^{2}(\frac{1}{5} x^{2}-\frac{2}{3} xy+\frac{1}{4} y^{2}) \) Simplify the expression by following steps: - step0: Simplify: \(\frac{2}{5}y^{2}\left(\frac{1}{5}x^{2}-\frac{2}{3}xy+\frac{1}{4}y^{2}\right)\) - step1: Rewrite the expression: \(\frac{2}{5}\left(\frac{1}{5}x^{2}-\frac{2}{3}xy+\frac{1}{4}y^{2}\right)y^{2}\) - step2: Multiply the expression: \(\frac{1}{150}\left(12x^{2}-40xy+15y^{2}\right)y^{2}\) - step3: Rearrange the terms: \(\frac{2}{25}x^{2}y^{2}-\frac{4}{15}xy^{3}+\frac{1}{10}y^{4}\) Expand the expression \( -\frac{1}{4} x^{2}y^{2}z(2x-3yz-5z^{2}) \) Simplify the expression by following steps: - step0: Multiply the terms: \(-\frac{1}{4}x^{2}y^{2}z\left(2x-3yz-5z^{2}\right)\) - step1: Apply the distributive property: \(-\frac{1}{4}x^{2}y^{2}z\times 2x-\left(-\frac{1}{4}x^{2}y^{2}z\times 3yz\right)-\left(-\frac{1}{4}x^{2}y^{2}z\times 5z^{2}\right)\) - step2: Multiply the terms: \(-\frac{1}{2}x^{3}y^{2}z-\left(-\frac{3}{4}x^{2}y^{3}z^{2}\right)-\left(-\frac{5}{4}x^{2}y^{2}z^{3}\right)\) - step3: Remove the parentheses: \(-\frac{1}{2}x^{3}y^{2}z+\frac{3}{4}x^{2}y^{3}z^{2}+\frac{5}{4}x^{2}y^{2}z^{3}\) Expand the expression \( 3m^{2}(m^{a+5}-4m^{a+3}-2m^{a+1}+5m^{a+4}) \) Simplify the expression by following steps: - step0: Multiply the terms: \(3m^{2}\left(m^{a+5}-4m^{a+3}-2m^{a+1}+5m^{a+4}\right)\) - step1: Use the distributive property: \(3m^{2}\times m^{a+5}+3m^{2}\left(-4m^{a+3}\right)+3m^{2}\left(-2m^{a+1}\right)+3m^{2}\times 5m^{a+4}\) - step2: Multiply the terms: \(3m^{a+7}+3m^{2}\left(-4m^{a+3}\right)+3m^{2}\left(-2m^{a+1}\right)+3m^{2}\times 5m^{a+4}\) - step3: Multiply the terms: \(3m^{a+7}-12m^{a+5}+3m^{2}\left(-2m^{a+1}\right)+3m^{2}\times 5m^{a+4}\) - step4: Multiply the terms: \(3m^{a+7}-12m^{a+5}-6m^{a+3}+3m^{2}\times 5m^{a+4}\) - step5: Multiply the terms: \(3m^{a+7}-12m^{a+5}-6m^{a+3}+15m^{a+6}\) Expand the expression \( 2x^{2}y(x^{a}y^{b}+x^{a-1}y^{b-1}-x^{a-2}y^{b+2}) \) Simplify the expression by following steps: - step0: Multiply the terms: \(2x^{2}y\left(x^{a}y^{b}+x^{a-1}y^{b-1}-x^{a-2}y^{b+2}\right)\) - step1: Use the distributive property: \(2x^{2}yx^{a}y^{b}+2x^{2}yx^{a-1}y^{b-1}+2x^{2}y\left(-x^{a-2}y^{b+2}\right)\) - step2: Multiply the terms: \(2x^{a+2}y^{b+1}+2x^{2}yx^{a-1}y^{b-1}+2x^{2}y\left(-x^{a-2}y^{b+2}\right)\) - step3: Multiply the terms: \(2x^{a+2}y^{b+1}+2x^{a+1}y^{b}+2x^{2}y\left(-x^{a-2}y^{b+2}\right)\) - step4: Multiply the terms: \(2x^{a+2}y^{b+1}+2x^{a+1}y^{b}-2x^{a}y^{b+3}\) Aquí están los resultados de las multiplicaciones solicitadas: c) \( mn^{3} \) por \( m^{3}-2mn-4n^{2} \): \[ m^{4}n^{3}-2m^{2}n^{4}-4mn^{5} \] d) \( 2x^{2}y \) por \( x^{a}y^{b}+x^{a-1}y^{b-1}-x^{a-2}y^{b+2} \): \[ 2x^{a+2}y^{b+1}+2x^{a+1}y^{b}-2x^{a}y^{b+3} \] e) \( \frac{1}{2} m^{3} n^{4} \) por \( \frac{1}{3} m^{4}-m^{2} n^{2}+\frac{1}{4} n^{2} \): \[ \frac{1}{6}m^{7}n^{4}-\frac{1}{2}m^{5}n^{6}+\frac{1}{8}n^{6}m^{3} \] f) \( \frac{2}{5} y^{2} \) por \( \frac{1}{5} x^{2}-\frac{2}{3} xy+\frac{1}{4} y^{2} \): \[ \frac{2}{25}x^{2}y^{2}-\frac{4}{15}xy^{3}+\frac{1}{10}y^{4} \] g) \( -2 x^{2} y \) por \( x^{3}+5 x^{2} y^{2}-3 y^{4} \): \[ -2x^{5}y-10x^{4}y^{3}+6x^{2}y^{5} \] h) \( 3 m^{2} \) por \( m^{a+5}-4 m^{a+3}-2 m^{a+1}+5 m^{a+4} \): \[ 3m^{a+7}-12m^{a+5}-6m^{a+3}+15m^{a+6} \] i) \( -\frac{1}{4} x^{2} y^{2} z \) por \( 2 x-3 y z-5 z^{2} \): \[ -\frac{1}{2}x^{3}y^{2}z+\frac{3}{4}x^{2}y^{3}z^{2}+\frac{5}{4}x^{2}y^{2}z^{3} \] Si necesitas más ayuda o explicaciones sobre algún paso, no dudes en preguntar.