Worksheet #5 on multiplying and dividing rational expressions, requiring factoring and simplification.

Let’s solve each problem step by step. We’ll multiply or divide the rational expressions, factor everything we can, and simplify.

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Problem 1:
$\frac{2a^2b}{b^2c} \cdot \frac{b}{a}$

Step 1: Multiply numerators and denominators:
Numerator: $2a^2b \cdot b = 2a^2b^2$
Denominator: $b^2c \cdot a = ab^2c$

So we have: $\frac{2a^2b^2}{ab^2c}$

Step 2: Cancel common factors:
- $a^2 / a = a$
- $b^2 / b^2 = 1$
Left with: $\frac{2a}{c}$

✔ Final Answer for #1: $\boxed{\frac{2a}{c}}$

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Problem 2:
$\frac{y^2 - 2y - 15}{4} \cdot \frac{8}{y + 3}$

Step 1: Factor numerator of first fraction:
$y^2 - 2y - 15 = (y - 5)(y + 3)$

Now expression is:
$\frac{(y - 5)(y + 3)}{4} \cdot \frac{8}{y + 3}$

Step 2: Cancel $(y + 3)$ top and bottom:
$\frac{(y - 5)}{4} \cdot \frac{8}{1}$

Step 3: Multiply:
$(y - 5) \cdot \frac{8}{4} = (y - 5) \cdot 2 = 2(y - 5)$

✔ Final Answer for #2: $\boxed{2(y - 5)}$

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Problem 3:
$\frac{x - 5}{6} \div \frac{2x - 10}{12}$

Step 1: Division → flip second fraction and multiply:
$\frac{x - 5}{6} \cdot \frac{12}{2x - 10}$

Step 2: Factor denominator of second fraction:
$2x - 10 = 2(x - 5)$

Now: $\frac{x - 5}{6} \cdot \frac{12}{2(x - 5)}$

Step 3: Cancel $(x - 5)$ top and bottom:
$\frac{1}{6} \cdot \frac{12}{2}$

Step 4: Simplify numbers:
$\frac{12}{6 \cdot 2} = \frac{12}{12} = 1$

✔ Final Answer for #3: $\boxed{1}$

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Problem 4:
$\frac{5n + 15}{4n + 8} \cdot \frac{2n + 4}{3n + 9}$

Step 1: Factor all parts:
- $5n + 15 = 5(n + 3)$
- $4n + 8 = 4(n + 2)$
- $2n + 4 = 2(n + 2)$
- $3n + 9 = 3(n + 3)$

Expression becomes:
$\frac{5(n + 3)}{4(n + 2)} \cdot \frac{2(n + 2)}{3(n + 3)}$

Step 2: Cancel common factors:
- $(n + 3)$ cancels
- $(n + 2)$ cancels
Left with: $\frac{5}{4} \cdot \frac{2}{3} = \frac{10}{12} = \frac{5}{6}$

✔ Final Answer for #4: $\boxed{\frac{5}{6}}$

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Problem 5:
$\frac{x^2 - 2x}{6} \div \frac{3x - 6}{x}$

Step 1: Flip second fraction and multiply:
$\frac{x^2 - 2x}{6} \cdot \frac{x}{3x - 6}$

Step 2: Factor:
- $x^2 - 2x = x(x - 2)$
- $3x - 6 = 3(x - 2)$

Expression: $\frac{x(x - 2)}{6} \cdot \frac{x}{3(x - 2)}$

Step 3: Cancel $(x - 2)$:
$\frac{x}{6} \cdot \frac{x}{3} = \frac{x^2}{18}$

✔ Final Answer for #5: $\boxed{\frac{x^2}{18}}$

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Problem 6:
$\frac{m^2 - 2m - 8}{8m + 24} \div \frac{2m - 8}{m^2 + 7m + 12}$

Step 1: Flip second fraction and multiply:
$\frac{m^2 - 2m - 8}{8m + 24} \cdot \frac{m^2 + 7m + 12}{2m - 8}$

Step 2: Factor everything:
- $m^2 - 2m - 8 = (m - 4)(m + 2)$
- $8m + 24 = 8(m + 3)$
- $m^2 + 7m + 12 = (m + 3)(m + 4)$
- $2m - 8 = 2(m - 4)$

Expression:
$\frac{(m - 4)(m + 2)}{8(m + 3)} \cdot \frac{(m + 3)(m + 4)}{2(m - 4)}$

Step 3: Cancel common factors:
- $(m - 4)$ cancels
- $(m + 3)$ cancels
Left with: $\frac{(m + 2)}{8} \cdot \frac{(m + 4)}{2} = \frac{(m + 2)(m + 4)}{16}$

✔ Final Answer for #6: $\boxed{\frac{(m + 2)(m + 4)}{16}}$

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Problem 7:
$\frac{x + 3}{10x + 20} \cdot \frac{x + 2}{x^2 + 4x + 3}$

Step 1: Factor:
- $10x + 20 = 10(x + 2)$
- $x^2 + 4x + 3 = (x + 1)(x + 3)$

Expression:
$\frac{x + 3}{10(x + 2)} \cdot \frac{x + 2}{(x + 1)(x + 3)}$

Step 2: Cancel:
- $(x + 3)$ cancels
- $(x + 2)$ cancels
Left with: $\frac{1}{10} \cdot \frac{1}{x + 1} = \frac{1}{10(x + 1)}$

✔ Final Answer for #7: $\boxed{\frac{1}{10(x + 1)}}$

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Problem 8:
$\frac{x^2 - x - 12}{x - 4} \div \frac{2x + 6}{x - 5}$

Step 1: Flip second fraction and multiply:
$\frac{x^2 - x - 12}{x - 4} \cdot \frac{x - 5}{2x + 6}$

Step 2: Factor:
- $x^2 - x - 12 = (x - 4)(x + 3)$
- $2x + 6 = 2(x + 3)$

Expression:
$\frac{(x - 4)(x + 3)}{x - 4} \cdot \frac{x - 5}{2(x + 3)}$

Step 3: Cancel:
- $(x - 4)$ cancels
- $(x + 3)$ cancels
Left with: $1 \cdot \frac{x - 5}{2} = \frac{x - 5}{2}$

✔ Final Answer for #8: $\boxed{\frac{x - 5}{2}}$

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Problem 9:
$\frac{x^2 - 5x - 6}{5x + 15} \div \frac{x^2 - 3x - 4}{7x + 21}$

Step 1: Flip and multiply:
$\frac{x^2 - 5x - 6}{5x + 15} \cdot \frac{7x + 21}{x^2 - 3x - 4}$

Step 2: Factor:
- $x^2 - 5x - 6 = (x - 6)(x + 1)$
- $5x + 15 = 5(x + 3)$
- $7x + 21 = 7(x + 3)$
- $x^2 - 3x - 4 = (x - 4)(x + 1)$

Expression:
$\frac{(x - 6)(x + 1)}{5(x + 3)} \cdot \frac{7(x + 3)}{(x - 4)(x + 1)}$

Step 3: Cancel:
- $(x + 1)$ cancels
- $(x + 3)$ cancels
Left with: $\frac{(x - 6)}{5} \cdot \frac{7}{(x - 4)} = \frac{7(x - 6)}{5(x - 4)}$

✔ Final Answer for #9: $\boxed{\frac{7(x - 6)}{5(x - 4)}}$

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Problem 10:
$\frac{24x^3}{25y^2} \cdot \frac{15y^2}{8x^2}$

Step 1: Multiply straight across:
Numerator: $24x^3 \cdot 15y^2 = 360x^3y^2$
Denominator: $25y^2 \cdot 8x^2 = 200x^2y^2$

Fraction: $\frac{360x^3y^2}{200x^2y^2}$

Step 2: Simplify coefficients:
$\frac{360}{200} = \frac{36}{20} = \frac{9}{5}$

Variables:
$x^3 / x^2 = x$, $y^2 / y^2 = 1$

Result: $\frac{9x}{5}$

✔ Final Answer for #10: $\boxed{\frac{9x}{5}}$

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Problem 11:
$\frac{6x - 18}{4x} \cdot \frac{x}{2x - 6}$

Step 1: Factor:
- $6x - 18 = 6(x - 3)$
- $2x - 6 = 2(x - 3)$

Expression:
$\frac{6(x - 3)}{4x} \cdot \frac{x}{2(x - 3)}$

Step 2: Cancel:
- $(x - 3)$ cancels
- $x$ cancels
Left with: $\frac{6}{4} \cdot \frac{1}{2} = \frac{6}{8} = \frac{3}{4}$

✔ Final Answer for #11: $\boxed{\frac{3}{4}}$

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Problem 12:
$\frac{3x + 12}{12x} \div \frac{x + 4}{48x^3}$

Step 1: Flip and multiply:
$\frac{3x + 12}{12x} \cdot \frac{48x^3}{x + 4}$

Step 2: Factor:
- $3x + 12 = 3(x + 4)$

Expression:
$\frac{3(x + 4)}{12x} \cdot \frac{48x^3}{x + 4}$

Step 3: Cancel:
- $(x + 4)$ cancels
Left with: $\frac{3}{12x} \cdot 48x^3 = \frac{3 \cdot 48x^3}{12x}$

Simplify numbers:
$\frac{144x^3}{12x} = 12x^2$

✔ Final Answer for #12: $\boxed{12x^2}$

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Final Answers:

1. $\boxed{\frac{2a}{c}}$
2. $\boxed{2(y - 5)}$
3. $\boxed{1}$
4. $\boxed{\frac{5}{6}}$
5. $\boxed{\frac{x^2}{18}}$
6. $\boxed{\frac{(m + 2)(m + 4)}{16}}$
7. $\boxed{\frac{1}{10(x + 1)}}$
8. $\boxed{\frac{x - 5}{2}}$
9. $\boxed{\frac{7(x - 6)}{5(x - 4)}}$
10. $\boxed{\frac{9x}{5}}$
11. $\boxed{\frac{3}{4}}$
12. $\boxed{12x^2}$