Simplify (50-2w^2)/(3w^2+9w-30)*(w^2+5w-14)/(6w-30)

$\frac{50 - 2 w^{2}}{3 w^{2} + 9 w - 30} \cdot \frac{w^{2} + 5 w - 14}{6 w - 30}$

Simplify the numerator.

Factor $2$ out of $50 - 2 w^{2}$ .

Factor $2$ out of $50$ .

$\frac{2 (25) - 2 w^{2}}{3 w^{2} + 9 w - 30} \cdot \frac{w^{2} + 5 w - 14}{6 w - 30}$

Factor $2$ out of $- 2 w^{2}$ .

$\frac{2 (25) + 2 (- w^{2})}{3 w^{2} + 9 w - 30} \cdot \frac{w^{2} + 5 w - 14}{6 w - 30}$

Factor $2$ out of $2 (25) + 2 (- w^{2})$ .

$\frac{2 (25 - w^{2})}{3 w^{2} + 9 w - 30} \cdot \frac{w^{2} + 5 w - 14}{6 w - 30}$

Rewrite $25$ as $5^{2}$ .

$\frac{2 (5^{2} - w^{2})}{3 w^{2} + 9 w - 30} \cdot \frac{w^{2} + 5 w - 14}{6 w - 30}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = w$ .

$\frac{2 (5 + w) (5 - w)}{3 w^{2} + 9 w - 30} \cdot \frac{w^{2} + 5 w - 14}{6 w - 30}$

Simplify the denominator.

Factor $3$ out of $3 w^{2} + 9 w - 30$ .

Factor $3$ out of $3 w^{2}$ .

$\frac{2 (5 + w) (5 - w)}{3 (w^{2}) + 9 w - 30} \cdot \frac{w^{2} + 5 w - 14}{6 w - 30}$

Factor $3$ out of $9 w$ .

$\frac{2 (5 + w) (5 - w)}{3 (w^{2}) + 3 (3 w) - 30} \cdot \frac{w^{2} + 5 w - 14}{6 w - 30}$

Factor $3$ out of $- 30$ .

$\frac{2 (5 + w) (5 - w)}{3 w^{2} + 3 (3 w) + 3 \cdot - 10} \cdot \frac{w^{2} + 5 w - 14}{6 w - 30}$

Factor $3$ out of $3 w^{2} + 3 (3 w)$ .

$\frac{2 (5 + w) (5 - w)}{3 (w^{2} + 3 w) + 3 \cdot - 10} \cdot \frac{w^{2} + 5 w - 14}{6 w - 30}$

Factor $3$ out of $3 (w^{2} + 3 w) + 3 \cdot - 10$ .

$\frac{2 (5 + w) (5 - w)}{3 (w^{2} + 3 w - 10)} \cdot \frac{w^{2} + 5 w - 14}{6 w - 30}$

Factor $w^{2} + 3 w - 10$ 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 $- 10$ and whose sum is $3$ .

$- 2, 5$

Write the factored form using these integers.

$\frac{2 (5 + w) (5 - w)}{3 ((w - 2) (w + 5))} \cdot \frac{w^{2} + 5 w - 14}{6 w - 30}$

$\frac{2 (5 + w) (5 - w)}{3 (w - 2) (w + 5)} \cdot \frac{w^{2} + 5 w - 14}{6 w - 30}$

Factor $w^{2} + 5 w - 14$ 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 $- 14$ and whose sum is $5$ .

$- 2, 7$

Write the factored form using these integers.

$\frac{2 (5 + w) (5 - w)}{3 (w - 2) (w + 5)} \cdot \frac{(w - 2) (w + 7)}{6 w - 30}$

Simplify terms.

Factor $6$ out of $6 w - 30$ .

Factor $6$ out of $6 w$ .

$\frac{2 (5 + w) (5 - w)}{3 (w - 2) (w + 5)} \cdot \frac{(w - 2) (w + 7)}{6 (w) - 30}$

Factor $6$ out of $- 30$ .

$\frac{2 (5 + w) (5 - w)}{3 (w - 2) (w + 5)} \cdot \frac{(w - 2) (w + 7)}{6 w + 6 \cdot - 5}$

Factor $6$ out of $6 w + 6 \cdot - 5$ .

$\frac{2 (5 + w) (5 - w)}{3 (w - 2) (w + 5)} \cdot \frac{(w - 2) (w + 7)}{6 (w - 5)}$

Cancel the common factor of $2$ .

Factor $2$ out of $2 (5 + w) (5 - w)$ .

$\frac{2 ((5 + w) (5 - w))}{3 (w - 2) (w + 5)} \cdot \frac{(w - 2) (w + 7)}{6 (w - 5)}$

Factor $2$ out of $6 (w - 5)$ .

$\frac{2 ((5 + w) (5 - w))}{3 (w - 2) (w + 5)} \cdot \frac{(w - 2) (w + 7)}{2 (3 (w - 5))}$

Cancel the common factor.

$\frac{2 ((5 + w) (5 - w))}{3 (w - 2) (w + 5)} \cdot \frac{(w - 2) (w + 7)}{2 (3 (w - 5))}$

Rewrite the expression.

$\frac{(5 + w) (5 - w)}{3 (w - 2) (w + 5)} \cdot \frac{(w - 2) (w + 7)}{3 (w - 5)}$

Cancel the common factor of $w - 2$ .

Factor $w - 2$ out of $3 (w - 2) (w + 5)$ .

$\frac{(5 + w) (5 - w)}{(w - 2) (3 (w + 5))} \cdot \frac{(w - 2) (w + 7)}{3 (w - 5)}$

Cancel the common factor.

$\frac{(5 + w) (5 - w)}{(w - 2) (3 (w + 5))} \cdot \frac{(w - 2) (w + 7)}{3 (w - 5)}$

Rewrite the expression.

$\frac{(5 + w) (5 - w)}{3 (w + 5)} \cdot \frac{w + 7}{3 (w - 5)}$

Multiply $\frac{(5 + w) (5 - w)}{3 (w + 5)}$ by $\frac{w + 7}{3 (w - 5)}$ .

$\frac{(5 + w) (5 - w) (w + 7)}{3 (w + 5) (3 (w - 5))}$

Multiply $3$ by $3$ .

$\frac{(5 + w) (5 - w) (w + 7)}{9 (w + 5) (w - 5)}$

Cancel the common factor of $5 + w$ and $w + 5$ .

Reorder terms.

$\frac{(w + 5) (5 - w) (w + 7)}{9 (w + 5) (w - 5)}$

Cancel the common factor.

$\frac{(w + 5) (5 - w) (w + 7)}{9 (w + 5) (w - 5)}$

Rewrite the expression.

$\frac{(5 - w) (w + 7)}{9 (w - 5)}$

Cancel the common factor of $5 - w$ and $w - 5$ .

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{(- 1 (- 5) - w) (w + 7)}{9 (w - 5)}$

Factor $- 1$ out of $- w$ .

$\frac{(- 1 (- 5) - (w)) (w + 7)}{9 (w - 5)}$

Factor $- 1$ out of $- 1 (- 5) - (w)$ .

$\frac{- 1 (- 5 + w) (w + 7)}{9 (w - 5)}$

Reorder terms.

$\frac{- 1 (w - 5) (w + 7)}{9 (w - 5)}$

Cancel the common factor.

$\frac{- 1 (w - 5) (w + 7)}{9 (w - 5)}$

Rewrite the expression.

$\frac{- 1 (w + 7)}{9}$

Move the negative in front of the fraction.

$- \frac{w + 7}{9}$

Simplify (50-2w^2)/(3w^2+9w-30)*(w^2+5w-14)/(6w-30)

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