\( \begin{array}{ll}\left(5 p^{2}+q^{4}\right)\left(p^{3}-2 q^{2}\right) & \text { (8) } \\ \text { 7) } & \left(4 b^{2}+5 c^{2}\right)\left(-b^{3}+2 c^{5}\right) \\ \text { 9) } & \text { (10) }\left(\frac{1}{2} y-\frac{1}{3} y^{2}\right) \\ \text { (11) }\left(\frac{1}{2} a+\frac{1}{3}\right)\left(a+\frac{4}{3}\right)\left(4 a-\frac{3}{2} b\right) & \text { (12) }\left(2 q^{2}+8 r^{-3}\right)\left(q^{2}+\frac{1}{2} r^{2}\right)\end{array} \)

Simplify the expression by following steps: - step0: Remove the parentheses: \(\left(\frac{1}{2}y-\frac{1}{3}y^{2}\right)\) - step1: Remove the parentheses: \(\frac{1}{2}y-\frac{1}{3}y^{2}\) Expand the expression \( (4 b^{2}+5 c^{2})(-b^{3}+2 c^{5}) \) Simplify the expression by following steps: - step0: Multiply the terms: \(\left(4b^{2}+5c^{2}\right)\left(-b^{3}+2c^{5}\right)\) - step1: Apply the distributive property: \(4b^{2}\left(-b^{3}\right)+4b^{2}\times 2c^{5}+5c^{2}\left(-b^{3}\right)+5c^{2}\times 2c^{5}\) - step2: Multiply the terms: \(-4b^{5}+8b^{2}c^{5}-5c^{2}b^{3}+10c^{7}\) Expand the expression \( (2 q^{2}+8 r^{-3})(q^{2}+\frac{1}{2} r^{2}) \) Simplify the expression by following steps: - step0: Multiply the terms: \(\left(2q^{2}+8r^{-3}\right)\left(q^{2}+\frac{1}{2}r^{2}\right)\) - step1: Rewrite the expression: \(\frac{2q^{2}r^{3}+8}{r^{3}}\times \left(q^{2}+\frac{1}{2}r^{2}\right)\) - step2: Rewrite the expression: \(\frac{2q^{2}r^{3}+8}{r^{3}}\times \frac{2q^{2}+r^{2}}{2}\) - step3: Rewrite the expression: \(\frac{2\left(q^{2}r^{3}+4\right)}{r^{3}}\times \frac{2q^{2}+r^{2}}{2}\) - step4: Reduce the fraction: \(\frac{q^{2}r^{3}+4}{r^{3}}\times \left(2q^{2}+r^{2}\right)\) - step5: Multiply the terms: \(\frac{\left(q^{2}r^{3}+4\right)\left(2q^{2}+r^{2}\right)}{r^{3}}\) - step6: Multiply the terms: \(\frac{2q^{4}r^{3}+q^{2}r^{5}+8q^{2}+4r^{2}}{r^{3}}\) Expand the expression \( (5 p^{2}+q^{4})(p^{3}-2 q^{2}) \) Simplify the expression by following steps: - step0: Multiply the terms: \(\left(5p^{2}+q^{4}\right)\left(p^{3}-2q^{2}\right)\) - step1: Apply the distributive property: \(5p^{2}\times p^{3}-5p^{2}\times 2q^{2}+q^{4}p^{3}-q^{4}\times 2q^{2}\) - step2: Multiply the terms: \(5p^{5}-10p^{2}q^{2}+q^{4}p^{3}-2q^{6}\) Expand the expression \( (\frac{1}{2} a+\frac{1}{3})(a+\frac{4}{3})(4 a-\frac{3}{2} b) \) Simplify the expression by following steps: - step0: Simplify: \(\left(\frac{1}{2}a+\frac{1}{3}\right)\left(a+\frac{4}{3}\right)\left(4a-\frac{3}{2}b\right)\) - step1: Multiply the expression: \(\frac{1}{36}\left(72a^{3}-27a^{2}b+144a^{2}-54ab+64a-24b\right)\) - step2: Rearrange the terms: \(2a^{3}-\frac{3}{4}a^{2}b+4a^{2}-\frac{3}{2}ab+\frac{16}{9}a-\frac{2}{3}b\) Here are the expanded forms of the given expressions: 1. **For** \( (5 p^{2}+q^{4})(p^{3}-2 q^{2}) \): \[ 5p^{5} - 10p^{2}q^{2} + q^{4}p^{3} - 2q^{6} \] 2. **For** \( (4 b^{2}+5 c^{2})(-b^{3}+2 c^{5}) \): \[ -4b^{5} + 8b^{2}c^{5} - 5c^{2}b^{3} + 10c^{7} \] 3. **For** \( \left(\frac{1}{2} y-\frac{1}{3} y^{2}\right) \): \[ \frac{1}{2}y - \frac{1}{3}y^{2} \] 4. **For** \( \left(\frac{1}{2} a+\frac{1}{3}\right)\left(a+\frac{4}{3}\right)\left(4 a-\frac{3}{2} b\right) \): \[ 2a^{3} - \frac{3}{4}a^{2}b + 4a^{2} - \frac{3}{2}ab + \frac{16}{9}a - \frac{2}{3}b \] 5. **For** \( \left(2 q^{2}+8 r^{-3}\right)\left(q^{2}+\frac{1}{2} r^{2}\right) \): \[ \frac{2q^{4}r^{3} + q^{2}r^{5} + 8q^{2} + 4r^{2}}{r^{3}} \] These expansions provide a clearer view of the polynomial expressions involved. If you need further assistance or explanations on any specific part, feel free to ask!