Trigonometry Examples

$\frac{16 s^{2} + 8 s t + t^{2}}{2 s^{2} - 5 s t - 3 t^{2}} \cdot \frac{\frac{2 s^{2} - 7 s t + 3 t^{2}}{t^{2} + 3 s t - 4 s^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 1

Factor using the perfect square rule.

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Step 1.1

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

$\frac{{(4 s)}^{2} + 8 s t + t^{2}}{2 s^{2} - 5 s t - 3 t^{2}} \cdot \frac{\frac{2 s^{2} - 7 s t + 3 t^{2}}{t^{2} + 3 s t - 4 s^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 s t = 2 \cdot (4 s) \cdot t$

Step 1.3

Rewrite the polynomial.

$\frac{{(4 s)}^{2} + 2 \cdot (4 s) \cdot t + t^{2}}{2 s^{2} - 5 s t - 3 t^{2}} \cdot \frac{\frac{2 s^{2} - 7 s t + 3 t^{2}}{t^{2} + 3 s t - 4 s^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 4 s$ and $b = t$ .

$\frac{{(4 s + t)}^{2}}{2 s^{2} - 5 s t - 3 t^{2}} \cdot \frac{\frac{2 s^{2} - 7 s t + 3 t^{2}}{t^{2} + 3 s t - 4 s^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 2

Factor by grouping.

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Step 2.1

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 $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = - 5$ .

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Step 2.1.1

Reorder terms.

$\frac{{(4 s + t)}^{2}}{2 s^{2} - 3 t^{2} - 5 s t} \cdot \frac{\frac{2 s^{2} - 7 s t + 3 t^{2}}{t^{2} + 3 s t - 4 s^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 2.1.2

Reorder $- 3 t^{2}$ and $- 5 s t$ .

$\frac{{(4 s + t)}^{2}}{2 s^{2} - 5 s t - 3 t^{2}} \cdot \frac{\frac{2 s^{2} - 7 s t + 3 t^{2}}{t^{2} + 3 s t - 4 s^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 2.1.3

Factor $- 5$ out of $- 5 s t$ .

$\frac{{(4 s + t)}^{2}}{2 s^{2} - 5 (s t) - 3 t^{2}} \cdot \frac{\frac{2 s^{2} - 7 s t + 3 t^{2}}{t^{2} + 3 s t - 4 s^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 2.1.4

Rewrite $- 5$ as $1$ plus $- 6$

$\frac{{(4 s + t)}^{2}}{2 s^{2} + (1 - 6) (s t) - 3 t^{2}} \cdot \frac{\frac{2 s^{2} - 7 s t + 3 t^{2}}{t^{2} + 3 s t - 4 s^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 2.1.5

Apply the distributive property.

$\frac{{(4 s + t)}^{2}}{2 s^{2} + 1 (s t) - 6 (s t) - 3 t^{2}} \cdot \frac{\frac{2 s^{2} - 7 s t + 3 t^{2}}{t^{2} + 3 s t - 4 s^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 2.1.6

Multiply $s t$ by $1$ .

$\frac{{(4 s + t)}^{2}}{2 s^{2} + s t - 6 (s t) - 3 t^{2}} \cdot \frac{\frac{2 s^{2} - 7 s t + 3 t^{2}}{t^{2} + 3 s t - 4 s^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 2.1.7

Move parentheses.

$\frac{{(4 s + t)}^{2}}{2 s^{2} + s t - 6 s t - 3 t^{2}} \cdot \frac{\frac{2 s^{2} - 7 s t + 3 t^{2}}{t^{2} + 3 s t - 4 s^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$\frac{{(4 s + t)}^{2}}{(2 s^{2} + s t) - 6 s t - 3 t^{2}} \cdot \frac{\frac{2 s^{2} - 7 s t + 3 t^{2}}{t^{2} + 3 s t - 4 s^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 2.2.2

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

$\frac{{(4 s + t)}^{2}}{s (2 s + t) - 3 t (2 s + t)} \cdot \frac{\frac{2 s^{2} - 7 s t + 3 t^{2}}{t^{2} + 3 s t - 4 s^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $2 s + t$ .

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{2 s^{2} - 7 s t + 3 t^{2}}{t^{2} + 3 s t - 4 s^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 3

Simplify the numerator.

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Step 3.1

Factor by grouping.

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Step 3.1.1

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 $a \cdot c = 2 \cdot 3 = 6$ and whose sum is $b = - 7$ .

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Step 3.1.1.1

Reorder terms.

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{2 s^{2} + 3 t^{2} - 7 s t}{t^{2} + 3 s t - 4 s^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 3.1.1.2

Reorder $3 t^{2}$ and $- 7 s t$ .

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{2 s^{2} - 7 s t + 3 t^{2}}{t^{2} + 3 s t - 4 s^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 3.1.1.3

Factor $- 7$ out of $- 7 s t$ .

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{2 s^{2} - 7 (s t) + 3 t^{2}}{t^{2} + 3 s t - 4 s^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 3.1.1.4

Rewrite $- 7$ as $- 1$ plus $- 6$

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{2 s^{2} + (- 1 - 6) (s t) + 3 t^{2}}{t^{2} + 3 s t - 4 s^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 3.1.1.5

Apply the distributive property.

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{2 s^{2} - 1 (s t) - 6 (s t) + 3 t^{2}}{t^{2} + 3 s t - 4 s^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 3.1.1.6

Move parentheses.

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{2 s^{2} - 1 s t - 6 s t + 3 t^{2}}{t^{2} + 3 s t - 4 s^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 3.1.2

Factor out the greatest common factor from each group.

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Step 3.1.2.1

Group the first two terms and the last two terms.

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s^{2} - 1 s t) - 6 s t + 3 t^{2}}{t^{2} + 3 s t - 4 s^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 3.1.2.2

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

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{s (2 s - 1 t) - 3 t (2 s - t)}{t^{2} + 3 s t - 4 s^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 3.1.3

Factor the polynomial by factoring out the greatest common factor, $2 s - 1 t$ .

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - 1 t) (s - 3 t)}{t^{2} + 3 s t - 4 s^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 3.2

Rewrite $- 1 t$ as $- t$ .

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{t^{2} + 3 s t - 4 s^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 4

Simplify the denominator.

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Step 4.1

Factor by grouping.

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Step 4.1.1

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 $a \cdot c = - 4 \cdot 1 = - 4$ and whose sum is $b = 3$ .

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Step 4.1.1.1

Reorder terms.

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{- 4 s^{2} + t^{2} + 3 s t}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 4.1.1.2

Reorder $t^{2}$ and $3 s t$ .

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{- 4 s^{2} + 3 s t + t^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 4.1.1.3

Factor $3$ out of $3 s t$ .

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{- 4 s^{2} + 3 (s t) + t^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 4.1.1.4

Rewrite $3$ as $- 1$ plus $4$

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{- 4 s^{2} + (- 1 + 4) (s t) + t^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 4.1.1.5

Apply the distributive property.

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{- 4 s^{2} - 1 (s t) + 4 (s t) + t^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 4.1.1.6

Move parentheses.

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{- 4 s^{2} - 1 s t + 4 s t + t^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 4.1.2

Factor out the greatest common factor from each group.

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Step 4.1.2.1

Group the first two terms and the last two terms.

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{(- 4 s^{2} - 1 s t) + 4 s t + t^{2}}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 4.1.2.2

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

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{s (- 4 s - 1 t) - t (- 4 s - t)}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 4.1.3

Factor the polynomial by factoring out the greatest common factor, $- 4 s - 1 t$ .

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{(- 4 s - 1 t) (s - t)}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 4.2

Rewrite $- 1 t$ as $- t$ .

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 5

Factor by grouping.

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Step 5.1

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 $a \cdot c = 8 \cdot - 1 = - 8$ and whose sum is $b = - 2$ .

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Step 5.1.1

Reorder terms.

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{\frac{8 s^{2} - t^{2} - 2 s t}{2 s^{2} + 3 s t + t^{2}}}$

Step 5.1.2

Reorder $- t^{2}$ and $- 2 s t$ .

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{\frac{8 s^{2} - 2 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 5.1.3

Factor $- 2$ out of $- 2 s t$ .

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{\frac{8 s^{2} - 2 (s t) - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 5.1.4

Rewrite $- 2$ as $2$ plus $- 4$

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{\frac{8 s^{2} + (2 - 4) (s t) - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 5.1.5

Apply the distributive property.

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{\frac{8 s^{2} + 2 (s t) - 4 (s t) - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 5.1.6

Move parentheses.

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{\frac{8 s^{2} + 2 s t - 4 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 5.2

Factor out the greatest common factor from each group.

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Step 5.2.1

Group the first two terms and the last two terms.

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{\frac{(8 s^{2} + 2 s t) - 4 s t - t^{2}}{2 s^{2} + 3 s t + t^{2}}}$

Step 5.2.2

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

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{\frac{2 s (4 s + t) - t (4 s + t)}{2 s^{2} + 3 s t + t^{2}}}$

Step 5.3

Factor the polynomial by factoring out the greatest common factor, $4 s + t$ .

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{\frac{(4 s + t) (2 s - t)}{2 s^{2} + 3 s t + t^{2}}}$

Step 6

Factor by grouping.

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Step 6.1

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 $a \cdot c = 2 \cdot 1 = 2$ and whose sum is $b = 3$ .

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Step 6.1.1

Reorder terms.

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{\frac{(4 s + t) (2 s - t)}{2 s^{2} + t^{2} + 3 s t}}$

Step 6.1.2

Reorder $t^{2}$ and $3 s t$ .

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{\frac{(4 s + t) (2 s - t)}{2 s^{2} + 3 s t + t^{2}}}$

Step 6.1.3

Factor $3$ out of $3 s t$ .

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{\frac{(4 s + t) (2 s - t)}{2 s^{2} + 3 (s t) + t^{2}}}$

Step 6.1.4

Rewrite $3$ as $1$ plus $2$

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{\frac{(4 s + t) (2 s - t)}{2 s^{2} + (1 + 2) (s t) + t^{2}}}$

Step 6.1.5

Apply the distributive property.

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{\frac{(4 s + t) (2 s - t)}{2 s^{2} + 1 (s t) + 2 (s t) + t^{2}}}$

Step 6.1.6

Multiply $s t$ by $1$ .

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{\frac{(4 s + t) (2 s - t)}{2 s^{2} + s t + 2 (s t) + t^{2}}}$

Step 6.1.7

Move parentheses.

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{\frac{(4 s + t) (2 s - t)}{2 s^{2} + s t + 2 s t + t^{2}}}$

Step 6.2

Factor out the greatest common factor from each group.

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Step 6.2.1

Group the first two terms and the last two terms.

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{\frac{(4 s + t) (2 s - t)}{(2 s^{2} + s t) + 2 s t + t^{2}}}$

Step 6.2.2

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

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{\frac{(4 s + t) (2 s - t)}{s (2 s + t) + t (2 s + t)}}$

Step 6.3

Factor the polynomial by factoring out the greatest common factor, $2 s + t$ .

$\frac{{(4 s + t)}^{2}}{(2 s + t) (s - 3 t)} \cdot \frac{\frac{(2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{\frac{(4 s + t) (2 s - t)}{(2 s + t) (s + t)}}$

Step 7

Combine fractions.

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Step 7.1

Combine.

$\frac{{(4 s + t)}^{2} \frac{(2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{(2 s + t) (s - 3 t) \frac{(4 s + t) (2 s - t)}{(2 s + t) (s + t)}}$

Step 7.2

Combine ${(4 s + t)}^{2}$ and $\frac{(2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}$ .

$\frac{\frac{{(4 s + t)}^{2} ((2 s - t) (s - 3 t))}{(- 4 s - t) (s - t)}}{(2 s + t) (s - 3 t) \frac{(4 s + t) (2 s - t)}{(2 s + t) (s + t)}}$

Step 8

Cancel the common factor of ${(4 s + t)}^{2}$ and $- 4 s - t$ .

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Step 8.1

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

$\frac{\frac{{(- (- 4 s) + t)}^{2} (2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{(2 s + t) (s - 3 t) \frac{(4 s + t) (2 s - t)}{(2 s + t) (s + t)}}$

Step 8.2

Factor $- 1$ out of $t$ .

$\frac{\frac{{(- (- 4 s) - 1 (- t))}^{2} (2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{(2 s + t) (s - 3 t) \frac{(4 s + t) (2 s - t)}{(2 s + t) (s + t)}}$

Step 8.3

Factor $- 1$ out of $- (- 4 s) - 1 (- t)$ .

$\frac{\frac{{(- (- 4 s - t))}^{2} (2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{(2 s + t) (s - 3 t) \frac{(4 s + t) (2 s - t)}{(2 s + t) (s + t)}}$

Step 8.4

Rewrite $- (- 4 s - t)$ as $- 1 (- 4 s - t)$ .

$\frac{\frac{{(- 1 (- 4 s - t))}^{2} (2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{(2 s + t) (s - 3 t) \frac{(4 s + t) (2 s - t)}{(2 s + t) (s + t)}}$

Step 8.5

Apply the product rule to $- 1 (- 4 s - t)$ .

$\frac{\frac{{(- 1)}^{2} {(- 4 s - t)}^{2} (2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{(2 s + t) (s - 3 t) \frac{(4 s + t) (2 s - t)}{(2 s + t) (s + t)}}$

Step 8.6

Raise $- 1$ to the power of $2$ .

$\frac{\frac{1 {(- 4 s - t)}^{2} (2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{(2 s + t) (s - 3 t) \frac{(4 s + t) (2 s - t)}{(2 s + t) (s + t)}}$

Step 8.7

Multiply ${(- 4 s - t)}^{2}$ by $1$ .

$\frac{\frac{{(- 4 s - t)}^{2} (2 s - t) (s - 3 t)}{(- 4 s - t) (s - t)}}{(2 s + t) (s - 3 t) \frac{(4 s + t) (2 s - t)}{(2 s + t) (s + t)}}$

Step 8.8

Factor $- 4 s - t$ out of ${(- 4 s - t)}^{2} (2 s - t) (s - 3 t)$ .

$\frac{\frac{(- 4 s - t) (((- 4 s - t) (2 s - t)) (s - 3 t))}{(- 4 s - t) (s - t)}}{(2 s + t) (s - 3 t) \frac{(4 s + t) (2 s - t)}{(2 s + t) (s + t)}}$

Step 8.9

Cancel the common factors.

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Step 8.9.1

Cancel the common factor.

$\frac{\frac{(- 4 s - t) (((- 4 s - t) (2 s - t)) (s - 3 t))}{(- 4 s - t) (s - t)}}{(2 s + t) (s - 3 t) \frac{(4 s + t) (2 s - t)}{(2 s + t) (s + t)}}$

Step 8.9.2

Rewrite the expression.

$\frac{\frac{((- 4 s - t) (2 s - t)) (s - 3 t)}{s - t}}{(2 s + t) (s - 3 t) \frac{(4 s + t) (2 s - t)}{(2 s + t) (s + t)}}$

$\frac{\frac{(- 4 s - t) (2 s - t) (s - 3 t)}{s - t}}{(2 s + t) (s - 3 t) \frac{(4 s + t) (2 s - t)}{(2 s + t) (s + t)}}$

Step 9

Multiply the numerator by the reciprocal of the denominator.

$\frac{(- 4 s - t) (2 s - t) (s - 3 t)}{s - t} \cdot \frac{1}{(2 s + t) (s - 3 t) \frac{(4 s + t) (2 s - t)}{(2 s + t) (s + t)}}$

Step 10

Cancel the common factor of $s - 3 t$ .

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Step 10.1

Factor $s - 3 t$ out of $(- 4 s - t) (2 s - t) (s - 3 t)$ .

$\frac{(s - 3 t) ((- 4 s - t) (2 s - t))}{s - t} \cdot \frac{1}{(2 s + t) (s - 3 t) \frac{(4 s + t) (2 s - t)}{(2 s + t) (s + t)}}$

Step 10.2

Factor $s - 3 t$ out of $(2 s + t) (s - 3 t) \frac{(4 s + t) (2 s - t)}{(2 s + t) (s + t)}$ .

$\frac{(s - 3 t) ((- 4 s - t) (2 s - t))}{s - t} \cdot \frac{1}{(s - 3 t) ((2 s + t) \frac{(4 s + t) (2 s - t)}{(2 s + t) (s + t)})}$

Step 10.3

Cancel the common factor.

$\frac{(s - 3 t) ((- 4 s - t) (2 s - t))}{s - t} \cdot \frac{1}{(s - 3 t) ((2 s + t) \frac{(4 s + t) (2 s - t)}{(2 s + t) (s + t)})}$

Step 10.4

Rewrite the expression.

$\frac{(- 4 s - t) (2 s - t)}{s - t} \cdot \frac{1}{(2 s + t) \frac{(4 s + t) (2 s - t)}{(2 s + t) (s + t)}}$

Step 11

Multiply $\frac{(- 4 s - t) (2 s - t)}{s - t}$ by $\frac{1}{(2 s + t) \frac{(4 s + t) (2 s - t)}{(2 s + t) (s + t)}}$ .

$\frac{(- 4 s - t) (2 s - t)}{(s - t) ((2 s + t) \frac{(4 s + t) (2 s - t)}{(2 s + t) (s + t)})}$

Step 12

Factor $\frac{(4 s + t) (2 s - t)}{(2 s + t) (s + t)}$ out of $\frac{(- 4 s - t) (2 s - t)}{(s - t) (2 s + t) \frac{(4 s + t) (2 s - t)}{(2 s + t) (s + t)}}$ .

$\frac{(2 s + t) (s + t)}{(4 s + t) (2 s - t)} \cdot \frac{(- 4 s - t) (2 s - t)}{(s - t) (2 s + t)}$

Step 13

Cancel the common factor of $2 s + t$ .

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Step 13.1

Factor $2 s + t$ out of $(s - t) (2 s + t)$ .

$\frac{(2 s + t) (s + t)}{(4 s + t) (2 s - t)} \cdot \frac{(- 4 s - t) (2 s - t)}{(2 s + t) (s - t)}$

Step 13.2

Cancel the common factor.

$\frac{(2 s + t) (s + t)}{(4 s + t) (2 s - t)} \cdot \frac{(- 4 s - t) (2 s - t)}{(2 s + t) (s - t)}$

Step 13.3

Rewrite the expression.

$\frac{s + t}{(4 s + t) (2 s - t)} \cdot \frac{(- 4 s - t) (2 s - t)}{s - t}$

Step 14

Cancel the common factor of $2 s - t$ .

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Step 14.1

Factor $2 s - t$ out of $(4 s + t) (2 s - t)$ .

$\frac{s + t}{(2 s - t) (4 s + t)} \cdot \frac{(- 4 s - t) (2 s - t)}{s - t}$

Step 14.2

Factor $2 s - t$ out of $(- 4 s - t) (2 s - t)$ .

$\frac{s + t}{(2 s - t) (4 s + t)} \cdot \frac{(2 s - t) (- 4 s - t)}{s - t}$

Step 14.3

Cancel the common factor.

$\frac{s + t}{(2 s - t) (4 s + t)} \cdot \frac{(2 s - t) (- 4 s - t)}{s - t}$

Step 14.4

Rewrite the expression.

$\frac{s + t}{4 s + t} \cdot \frac{- 4 s - t}{s - t}$

Step 15

Multiply $\frac{s + t}{4 s + t}$ by $\frac{- 4 s - t}{s - t}$ .

$\frac{(s + t) (- 4 s - t)}{(4 s + t) (s - t)}$

Step 16

Cancel the common factor of $- 4 s - t$ and $4 s + t$ .

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Step 16.1

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

$\frac{(s + t) (- (4 s) - t)}{(4 s + t) (s - t)}$

Step 16.2

Factor $- 1$ out of $- t$ .

$\frac{(s + t) (- (4 s) - (t))}{(4 s + t) (s - t)}$

Step 16.3

Factor $- 1$ out of $- (4 s) - (t)$ .

$\frac{(s + t) (- (4 s + t))}{(4 s + t) (s - t)}$

Step 16.4

Rewrite $- (4 s + t)$ as $- 1 (4 s + t)$ .

$\frac{(s + t) (- 1 (4 s + t))}{(4 s + t) (s - t)}$

Step 16.5

Cancel the common factor.

$\frac{(s + t) (- 1 (4 s + t))}{(4 s + t) (s - t)}$

Step 16.6

Rewrite the expression.

$\frac{(s + t) \cdot (- 1)}{s - t}$

Step 17

Simplify the expression.

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Step 17.1

Move $- 1$ to the left of $s + t$ .

$\frac{- 1 \cdot (s + t)}{s - t}$

Step 17.2

Move the negative in front of the fraction.

$- \frac{s + t}{s - t}$