How to Multiply and Simplify the Rational Expressions (c+1)/(c²-2c-8) and (3c+5)/(c²-3c-4)

Question

Multiply the rational expressions: $\frac{c+1}{c^2-2c-8} \cdot \frac{3c+5}{c^2-3c-4}$

Accepted Answer

$\frac{3c+5}{(c-4)(c-4)(c+1)}$ or $\frac{3c+5}{(c-4)^2(c+1)}$, with restrictions $c \neq 4, -2, -1$

Answer

Step 1: Factor the quadratic denominators. For $c^2-2c-8$, find two numbers that multiply to -8 and add to -2: -4 and 2. So it factors to $(c-4)(c+2)$. For $c^2-3c-4$, find two numbers that multiply to -4 and add to -3: -4 and 1. So it factors to $(c-4)(c+1)$. Step 2: Rewrite the original expression with factored denominators: $\frac{c+1}{(c-4)(c+2)} \cdot \frac{3c+5}{(c-4)(c+1)}$ Step 3: Identify and cancel common factors in the numerator and denominator. The $(c+1)$ term appears in both the numerator and denominator, so they cancel out (note: $c \neq -1$ to avoid division by zero). Step 4: Multiply the remaining terms in the numerators and denominators. The numerator becomes $1 \cdot (3c+5) = 3c+5$. The denominator becomes $(c-4)(c+2) \cdot (c-4) = (c-4)^2(c+2)$. Step 5: State the restrictions on the variable: $c$ cannot be 4, -2, or -1, as these values would make the original denominators equal to zero.