Simplify (16w^2-28w-44)/(-16w^2+121)

$\frac{16 w^{2} - 28 w - 44}{- 16 w^{2} + 121}$

Step 1

Simplify the numerator.

Factor $4$ out of $16 w^{2} - 28 w - 44$ .

Factor $4$ out of $16 w^{2}$ .

$\frac{4 (4 w^{2}) - 28 w - 44}{- 16 w^{2} + 121}$

Factor $4$ out of $- 28 w$ .

$\frac{4 (4 w^{2}) + 4 (- 7 w) - 44}{- 16 w^{2} + 121}$

Factor $4$ out of $- 44$ .

$\frac{4 (4 w^{2}) + 4 (- 7 w) + 4 (- 11)}{- 16 w^{2} + 121}$

Factor $4$ out of $4 (4 w^{2}) + 4 (- 7 w)$ .

$\frac{4 (4 w^{2} - 7 w) + 4 (- 11)}{- 16 w^{2} + 121}$

Factor $4$ out of $4 (4 w^{2} - 7 w) + 4 (- 11)$ .

$\frac{4 (4 w^{2} - 7 w - 11)}{- 16 w^{2} + 121}$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot - 11 = - 44$ and whose sum is $b = - 7$ .

Factor $- 7$ out of $- 7 w$ .

$\frac{4 (4 w^{2} - 7 (w) - 11)}{- 16 w^{2} + 121}$

Rewrite $- 7$ as $4$ plus $- 11$

$\frac{4 (4 w^{2} + (4 - 11) w - 11)}{- 16 w^{2} + 121}$

Apply the distributive property.

$\frac{4 (4 w^{2} + 4 w - 11 w - 11)}{- 16 w^{2} + 121}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{4 ((4 w^{2} + 4 w) - 11 w - 11)}{- 16 w^{2} + 121}$

Factor out the greatest common factor (GCF) from each group.

$\frac{4 (4 w (w + 1) - 11 (w + 1))}{- 16 w^{2} + 121}$

Factor the polynomial by factoring out the greatest common factor, $w + 1$ .

$\frac{4 ((w + 1) (4 w - 11))}{- 16 w^{2} + 121}$

$\frac{4 (w + 1) (4 w - 11)}{- 16 w^{2} + 121}$

Step 2

Simplify the denominator.

Rewrite $121$ as $11^{2}$ .

$\frac{4 (w + 1) (4 w - 11)}{- 16 w^{2} + 11^{2}}$

Rewrite $16 w^{2}$ as ${(4 w)}^{2}$ .

$\frac{4 (w + 1) (4 w - 11)}{- {(4 w)}^{2} + 11^{2}}$

Reorder $- {(4 w)}^{2}$ and $11^{2}$ .

$\frac{4 (w + 1) (4 w - 11)}{11^{2} - {(4 w)}^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 11$ and $b = 4 w$ .

$\frac{4 (w + 1) (4 w - 11)}{(11 + 4 w) (11 - (4 w))}$

Multiply $4$ by $- 1$ .

$\frac{4 (w + 1) (4 w - 11)}{(11 + 4 w) (11 - 4 w)}$

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of $4 w - 11$ and $11 - 4 w$ .

Factor $- 1$ out of $4 w$ .

$\frac{4 (w + 1) (- (- 4 w) - 11)}{(11 + 4 w) (11 - 4 w)}$

Rewrite $- 11$ as $- 1 (11)$ .

$\frac{4 (w + 1) (- (- 4 w) - 1 (11))}{(11 + 4 w) (11 - 4 w)}$

Factor $- 1$ out of $- (- 4 w) - 1 (11)$ .

$\frac{4 (w + 1) (- (- 4 w + 11))}{(11 + 4 w) (11 - 4 w)}$

Rewrite $- (- 4 w + 11)$ as $- 1 (- 4 w + 11)$ .

$\frac{4 (w + 1) (- 1 (- 4 w + 11))}{(11 + 4 w) (11 - 4 w)}$

Reorder terms.

$\frac{4 (w + 1) (- 1 (- 4 w + 11))}{(11 + 4 w) (- 4 w + 11)}$

Cancel the common factor.

$\frac{4 (w + 1) (- 1 (- 4 w + 11))}{(11 + 4 w) (- 4 w + 11)}$

Rewrite the expression.

$\frac{4 (w + 1) \cdot (- 1)}{11 + 4 w}$

Simplify the expression.

Multiply $- 1$ by $4$ .

$\frac{- 4 (w + 1)}{11 + 4 w}$

Move the negative in front of the fraction.

$- \frac{4 (w + 1)}{11 + 4 w}$

Simplify (16w^2-28w-44)/(-16w^2+121)

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