Question

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divide. state the quotient in simplest form, and express the domain of the expression.
$\frac{y+3}{y+1} div \frac{y^2+4y+3}{-5-5y}$
$\bigcirc -\frac{y+1}{5}, y
eq -1$
$\bigcirc -\frac{5}{y+1}, y
eq -1$
$\bigcirc -\frac{y+1}{5}, y
eq 1$
$\bigcirc -\frac{5}{y+1}, y
eq 1$

Explanation:

Step1: Rewrite division as multiplication

$\frac{y+3}{y+1} \div \frac{y^2+4y+3}{-5-5y} = \frac{y+3}{y+1} \times \frac{-5-5y}{y^2+4y+3}$

Step2: Factor all expressions

$\frac{y+3}{y+1} \times \frac{-5(y+1)}{(y+1)(y+3)}$

Step3: Cancel common factors

$\frac{\cancel{y+3}}{\cancel{y+1}} \times \frac{-5\cancel{(y+1)}}{(y+1)\cancel{(y+3)}} = -\frac{5}{y+1}$

Step4: Find domain restrictions

Denominators cannot be zero: $y+1
eq 0 \implies y
eq -1$; also, original divisor cannot be zero: $y^2+4y+3
eq 0$ (already covered by $y
eq -1$) and $-5-5y
eq 0$ (also covered by $y
eq -1$).

Answer:

$-\frac{5}{y+1}, y
eq -1$