Simplify ((y+3)/(y+1))÷((y^2+4y+3)/(-5-5y))

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

To divide by a fraction, multiply by its reciprocal.

$\frac{y + 3}{y + 1} \cdot \frac{- 5 - 5 y}{y^{2} + 4 y + 3}$

Factor $5$ out of $- 5 - 5 y$ .

Factor $5$ out of $- 5$ .

$\frac{y + 3}{y + 1} \cdot \frac{5 (- 1) - 5 y}{y^{2} + 4 y + 3}$

Factor $5$ out of $- 5 y$ .

$\frac{y + 3}{y + 1} \cdot \frac{5 (- 1) + 5 (- y)}{y^{2} + 4 y + 3}$

Factor $5$ out of $5 (- 1) + 5 (- y)$ .

$\frac{y + 3}{y + 1} \cdot \frac{5 (- 1 - y)}{y^{2} + 4 y + 3}$

Factor $y^{2} + 4 y + 3$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $3$ and whose sum is $4$ .

$1, 3$

Write the factored form using these integers.

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

Simplify terms.

Cancel the common factor of $y + 3$ .

Factor $y + 3$ out of $(y + 1) (y + 3)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{1}{y + 1}$ by $\frac{5 (- 1 - y)}{y + 1}$ .

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

Raise $y + 1$ to the power of $1$ .

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

Raise $y + 1$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Reduce the expression by cancelling the common factors.

Add $1$ and $1$ .

$\frac{5 (- 1 - y)}{{(y + 1)}^{2}}$

Cancel the common factor of $- 1 - y$ and ${(y + 1)}^{2}$ .

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{5 (- 1 (1) - y)}{{(y + 1)}^{2}}$

Factor $- 1$ out of $- y$ .

$\frac{5 (- 1 (1) - (y))}{{(y + 1)}^{2}}$

Factor $- 1$ out of $- 1 (1) - (y)$ .

$\frac{5 (- 1 (1 + y))}{{(y + 1)}^{2}}$

Reorder terms.

$\frac{5 (- 1 (y + 1))}{{(y + 1)}^{2}}$

Factor $y + 1$ out of $5 (- 1 (y + 1))$ .

$\frac{(y + 1) (5 \cdot (- 1))}{{(y + 1)}^{2}}$

Cancel the common factors.

Factor $y + 1$ out of ${(y + 1)}^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $5$ by $- 1$ .

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

Move the negative in front of the fraction.

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

Simplify ((y+3)/(y+1))÷((y^2+4y+3)/(-5-5y))

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