Basic Math Examples

$\frac{4 t}{2 t^{2} + t - 36} + \frac{3}{16 - 4 t} - \frac{8}{2 t + 9}$

Step 1

Simplify each term.

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Step 1.1

Factor by grouping.

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Step 1.1.1

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 $a \cdot c = 2 \cdot - 36 = - 72$ and whose sum is $b = 1$ .

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Step 1.1.1.1

Multiply by $1$ .

$\frac{4 t}{2 t^{2} + 1 t - 36} + \frac{3}{16 - 4 t} - \frac{8}{2 t + 9}$

Step 1.1.1.2

Rewrite $1$ as $- 8$ plus $9$

$\frac{4 t}{2 t^{2} + (- 8 + 9) t - 36} + \frac{3}{16 - 4 t} - \frac{8}{2 t + 9}$

Step 1.1.1.3

Apply the distributive property.

$\frac{4 t}{2 t^{2} - 8 t + 9 t - 36} + \frac{3}{16 - 4 t} - \frac{8}{2 t + 9}$

Step 1.1.2

Factor out the greatest common factor from each group.

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Step 1.1.2.1

Group the first two terms and the last two terms.

$\frac{4 t}{(2 t^{2} - 8 t) + 9 t - 36} + \frac{3}{16 - 4 t} - \frac{8}{2 t + 9}$

Step 1.1.2.2

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

$\frac{4 t}{2 t (t - 4) + 9 (t - 4)} + \frac{3}{16 - 4 t} - \frac{8}{2 t + 9}$

Step 1.1.3

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

$\frac{4 t}{(t - 4) (2 t + 9)} + \frac{3}{16 - 4 t} - \frac{8}{2 t + 9}$

Step 1.2

Factor $4$ out of $16 - 4 t$ .

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Step 1.2.1

Factor $4$ out of $16$ .

$\frac{4 t}{(t - 4) (2 t + 9)} + \frac{3}{4 (4) - 4 t} - \frac{8}{2 t + 9}$

Step 1.2.2

Factor $4$ out of $- 4 t$ .

$\frac{4 t}{(t - 4) (2 t + 9)} + \frac{3}{4 (4) + 4 (- t)} - \frac{8}{2 t + 9}$

Step 1.2.3

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

$\frac{4 t}{(t - 4) (2 t + 9)} + \frac{3}{4 (4 - t)} - \frac{8}{2 t + 9}$

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{4 t}{(t - 4) (2 t + 9)}$ by $\frac{4 (4 - t)}{4 (4 - t)}$ .

$\frac{4 t}{(t - 4) (2 t + 9)} \cdot \frac{4 (4 - t)}{4 (4 - t)} + \frac{3}{4 (4 - t)} - \frac{8}{2 t + 9}$

Step 2.2

Multiply $\frac{4 t}{(t - 4) (2 t + 9)}$ by $\frac{4 (4 - t)}{4 (4 - t)}$ .

$\frac{4 t (4 (4 - t))}{(t - 4) (2 t + 9) (4 (4 - t))} + \frac{3}{4 (4 - t)} - \frac{8}{2 t + 9}$

Step 2.3

Multiply $\frac{3}{4 (4 - t)}$ by $\frac{(t - 4) (2 t + 9)}{(t - 4) (2 t + 9)}$ .

$\frac{4 t (4 (4 - t))}{(t - 4) (2 t + 9) (4 (4 - t))} + \frac{3}{4 (4 - t)} \cdot \frac{(t - 4) (2 t + 9)}{(t - 4) (2 t + 9)} - \frac{8}{2 t + 9}$

Step 2.4

Multiply $\frac{3}{4 (4 - t)}$ by $\frac{(t - 4) (2 t + 9)}{(t - 4) (2 t + 9)}$ .

$\frac{4 t (4 (4 - t))}{(t - 4) (2 t + 9) (4 (4 - t))} + \frac{3 ((t - 4) (2 t + 9))}{4 (4 - t) ((t - 4) (2 t + 9))} - \frac{8}{2 t + 9}$

Step 2.5

Multiply $\frac{8}{2 t + 9}$ by $\frac{- 4 {(- t + 4)}^{2}}{- 4 {(- t + 4)}^{2}}$ .

$\frac{4 t (4 (4 - t))}{(t - 4) (2 t + 9) (4 (4 - t))} + \frac{3 ((t - 4) (2 t + 9))}{4 (4 - t) ((t - 4) (2 t + 9))} - (\frac{8}{2 t + 9} \cdot \frac{- 4 {(- t + 4)}^{2}}{- 4 {(- t + 4)}^{2}})$

Step 2.6

Multiply $\frac{8}{2 t + 9}$ by $\frac{- 4 {(- t + 4)}^{2}}{- 4 {(- t + 4)}^{2}}$ .

$\frac{4 t (4 (4 - t))}{(t - 4) (2 t + 9) (4 (4 - t))} + \frac{3 ((t - 4) (2 t + 9))}{4 (4 - t) ((t - 4) (2 t + 9))} - \frac{8 (- 4 {(- t + 4)}^{2})}{(2 t + 9) (- 4 {(- t + 4)}^{2})}$

Step 2.7

Reorder the factors of $(t - 4) (2 t + 9) (4 (4 - t))$ .

$\frac{4 t (4 (4 - t))}{4 (2 t + 9) (4 - t) (t - 4)} + \frac{3 ((t - 4) (2 t + 9))}{4 (4 - t) ((t - 4) (2 t + 9))} - \frac{8 (- 4 {(- t + 4)}^{2})}{(2 t + 9) (- 4 {(- t + 4)}^{2})}$

Step 2.8

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

$\frac{4 t (4 (4 - t))}{4 (2 t + 9) (- 1 (- 4) - t) (t - 4)} + \frac{3 ((t - 4) (2 t + 9))}{4 (4 - t) ((t - 4) (2 t + 9))} - \frac{8 (- 4 {(- t + 4)}^{2})}{(2 t + 9) (- 4 {(- t + 4)}^{2})}$

Step 2.9

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

$\frac{4 t (4 (4 - t))}{4 (2 t + 9) (- 1 (- 4) - (t)) (t - 4)} + \frac{3 ((t - 4) (2 t + 9))}{4 (4 - t) ((t - 4) (2 t + 9))} - \frac{8 (- 4 {(- t + 4)}^{2})}{(2 t + 9) (- 4 {(- t + 4)}^{2})}$

Step 2.10

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

$\frac{4 t (4 (4 - t))}{4 (2 t + 9) (- 1 (- 4 + t)) (t - 4)} + \frac{3 ((t - 4) (2 t + 9))}{4 (4 - t) ((t - 4) (2 t + 9))} - \frac{8 (- 4 {(- t + 4)}^{2})}{(2 t + 9) (- 4 {(- t + 4)}^{2})}$

Step 2.11

Reorder terms.

$\frac{4 t (4 (4 - t))}{4 (2 t + 9) \cdot - 1 (t - 4) (t - 4)} + \frac{3 ((t - 4) (2 t + 9))}{4 (4 - t) ((t - 4) (2 t + 9))} - \frac{8 (- 4 {(- t + 4)}^{2})}{(2 t + 9) (- 4 {(- t + 4)}^{2})}$

Step 2.12

Raise $t - 4$ to the power of $1$ .

$\frac{4 t (4 (4 - t))}{4 (2 t + 9) \cdot - 1 ({(t - 4)}^{1} (t - 4))} + \frac{3 ((t - 4) (2 t + 9))}{4 (4 - t) ((t - 4) (2 t + 9))} - \frac{8 (- 4 {(- t + 4)}^{2})}{(2 t + 9) (- 4 {(- t + 4)}^{2})}$

Step 2.13

Raise $t - 4$ to the power of $1$ .

$\frac{4 t (4 (4 - t))}{4 (2 t + 9) \cdot - 1 ({(t - 4)}^{1} {(t - 4)}^{1})} + \frac{3 ((t - 4) (2 t + 9))}{4 (4 - t) ((t - 4) (2 t + 9))} - \frac{8 (- 4 {(- t + 4)}^{2})}{(2 t + 9) (- 4 {(- t + 4)}^{2})}$

Step 2.14

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

$\frac{4 t (4 (4 - t))}{4 (2 t + 9) \cdot - 1 {(t - 4)}^{1 + 1}} + \frac{3 ((t - 4) (2 t + 9))}{4 (4 - t) ((t - 4) (2 t + 9))} - \frac{8 (- 4 {(- t + 4)}^{2})}{(2 t + 9) (- 4 {(- t + 4)}^{2})}$

Step 2.15

Add $1$ and $1$ .

$\frac{4 t (4 (4 - t))}{4 (2 t + 9) \cdot - 1 {(t - 4)}^{2}} + \frac{3 ((t - 4) (2 t + 9))}{4 (4 - t) ((t - 4) (2 t + 9))} - \frac{8 (- 4 {(- t + 4)}^{2})}{(2 t + 9) (- 4 {(- t + 4)}^{2})}$

Step 2.16

Multiply $- 1$ by $4$ .

$\frac{4 t (4 (4 - t))}{- 4 (2 t + 9) {(t - 4)}^{2}} + \frac{3 ((t - 4) (2 t + 9))}{4 (4 - t) ((t - 4) (2 t + 9))} - \frac{8 (- 4 {(- t + 4)}^{2})}{(2 t + 9) (- 4 {(- t + 4)}^{2})}$

Step 2.17

Reorder the factors of $4 (4 - t) ((t - 4) (2 t + 9))$ .

$\frac{4 t (4 (4 - t))}{- 4 (2 t + 9) {(t - 4)}^{2}} + \frac{3 ((t - 4) (2 t + 9))}{4 (2 t + 9) (4 - t) (t - 4)} - \frac{8 (- 4 {(- t + 4)}^{2})}{(2 t + 9) (- 4 {(- t + 4)}^{2})}$

Step 2.18

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

$\frac{4 t (4 (4 - t))}{- 4 (2 t + 9) {(t - 4)}^{2}} + \frac{3 ((t - 4) (2 t + 9))}{4 (2 t + 9) (- 1 (- 4) - t) (t - 4)} - \frac{8 (- 4 {(- t + 4)}^{2})}{(2 t + 9) (- 4 {(- t + 4)}^{2})}$

Step 2.19

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

$\frac{4 t (4 (4 - t))}{- 4 (2 t + 9) {(t - 4)}^{2}} + \frac{3 ((t - 4) (2 t + 9))}{4 (2 t + 9) (- 1 (- 4) - (t)) (t - 4)} - \frac{8 (- 4 {(- t + 4)}^{2})}{(2 t + 9) (- 4 {(- t + 4)}^{2})}$

Step 2.20

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

$\frac{4 t (4 (4 - t))}{- 4 (2 t + 9) {(t - 4)}^{2}} + \frac{3 ((t - 4) (2 t + 9))}{4 (2 t + 9) (- 1 (- 4 + t)) (t - 4)} - \frac{8 (- 4 {(- t + 4)}^{2})}{(2 t + 9) (- 4 {(- t + 4)}^{2})}$

Step 2.21

Reorder terms.

$\frac{4 t (4 (4 - t))}{- 4 (2 t + 9) {(t - 4)}^{2}} + \frac{3 ((t - 4) (2 t + 9))}{4 (2 t + 9) \cdot - 1 (t - 4) (t - 4)} - \frac{8 (- 4 {(- t + 4)}^{2})}{(2 t + 9) (- 4 {(- t + 4)}^{2})}$

Step 2.22

Raise $t - 4$ to the power of $1$ .

$\frac{4 t (4 (4 - t))}{- 4 (2 t + 9) {(t - 4)}^{2}} + \frac{3 ((t - 4) (2 t + 9))}{4 (2 t + 9) \cdot - 1 ({(t - 4)}^{1} (t - 4))} - \frac{8 (- 4 {(- t + 4)}^{2})}{(2 t + 9) (- 4 {(- t + 4)}^{2})}$

Step 2.23

Raise $t - 4$ to the power of $1$ .

$\frac{4 t (4 (4 - t))}{- 4 (2 t + 9) {(t - 4)}^{2}} + \frac{3 ((t - 4) (2 t + 9))}{4 (2 t + 9) \cdot - 1 ({(t - 4)}^{1} {(t - 4)}^{1})} - \frac{8 (- 4 {(- t + 4)}^{2})}{(2 t + 9) (- 4 {(- t + 4)}^{2})}$

Step 2.24

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

$\frac{4 t (4 (4 - t))}{- 4 (2 t + 9) {(t - 4)}^{2}} + \frac{3 ((t - 4) (2 t + 9))}{4 (2 t + 9) \cdot - 1 {(t - 4)}^{1 + 1}} - \frac{8 (- 4 {(- t + 4)}^{2})}{(2 t + 9) (- 4 {(- t + 4)}^{2})}$

Step 2.25

Add $1$ and $1$ .

$\frac{4 t (4 (4 - t))}{- 4 (2 t + 9) {(t - 4)}^{2}} + \frac{3 ((t - 4) (2 t + 9))}{4 (2 t + 9) \cdot - 1 {(t - 4)}^{2}} - \frac{8 (- 4 {(- t + 4)}^{2})}{(2 t + 9) (- 4 {(- t + 4)}^{2})}$

Step 2.26

Multiply $- 1$ by $4$ .

$\frac{4 t (4 (4 - t))}{- 4 (2 t + 9) {(t - 4)}^{2}} + \frac{3 ((t - 4) (2 t + 9))}{- 4 (2 t + 9) {(t - 4)}^{2}} - \frac{8 (- 4 {(- t + 4)}^{2})}{(2 t + 9) (- 4 {(- t + 4)}^{2})}$

Step 2.27

Reorder the factors of $(2 t + 9) (- 4 {(- t + 4)}^{2})$ .

$\frac{4 t (4 (4 - t))}{- 4 (2 t + 9) {(t - 4)}^{2}} + \frac{3 ((t - 4) (2 t + 9))}{- 4 (2 t + 9) {(t - 4)}^{2}} - \frac{8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 3

Combine the numerators over the common denominator.

$\frac{4 t (4 (4 - t)) + 3 ((t - 4) (2 t + 9))}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 4

Simplify each term.

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Step 4.1

Apply the distributive property.

$\frac{4 t (4 \cdot 4 + 4 (- t)) + 3 ((t - 4) (2 t + 9))}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 4.2

Multiply $4$ by $4$ .

$\frac{4 t (16 + 4 (- t)) + 3 ((t - 4) (2 t + 9))}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 4.3

Multiply $- 1$ by $4$ .

$\frac{4 t (16 - 4 t) + 3 ((t - 4) (2 t + 9))}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 4.4

Apply the distributive property.

$\frac{4 t \cdot 16 + 4 t (- 4 t) + 3 ((t - 4) (2 t + 9))}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 4.5

Multiply $16$ by $4$ .

$\frac{64 t + 4 t (- 4 t) + 3 ((t - 4) (2 t + 9))}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 4.6

Rewrite using the commutative property of multiplication.

$\frac{64 t + 4 \cdot - 4 t \cdot t + 3 ((t - 4) (2 t + 9))}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 4.7

Simplify each term.

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Step 4.7.1

Multiply $t$ by $t$ by adding the exponents.

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Step 4.7.1.1

Move $t$ .

$\frac{64 t + 4 \cdot - 4 (t \cdot t) + 3 ((t - 4) (2 t + 9))}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 4.7.1.2

Multiply $t$ by $t$ .

$\frac{64 t + 4 \cdot - 4 t^{2} + 3 ((t - 4) (2 t + 9))}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 4.7.2

Multiply $4$ by $- 4$ .

$\frac{64 t - 16 t^{2} + 3 ((t - 4) (2 t + 9))}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 4.8

Expand $(t - 4) (2 t + 9)$ using the FOIL Method.

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Step 4.8.1

Apply the distributive property.

$\frac{64 t - 16 t^{2} + 3 (t (2 t + 9) - 4 (2 t + 9))}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 4.8.2

Apply the distributive property.

$\frac{64 t - 16 t^{2} + 3 (t (2 t) + t \cdot 9 - 4 (2 t + 9))}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 4.8.3

Apply the distributive property.

$\frac{64 t - 16 t^{2} + 3 (t (2 t) + t \cdot 9 - 4 (2 t) - 4 \cdot 9)}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 4.9

Simplify and combine like terms.

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Step 4.9.1

Simplify each term.

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Step 4.9.1.1

Rewrite using the commutative property of multiplication.

$\frac{64 t - 16 t^{2} + 3 (2 t \cdot t + t \cdot 9 - 4 (2 t) - 4 \cdot 9)}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 4.9.1.2

Multiply $t$ by $t$ by adding the exponents.

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Step 4.9.1.2.1

Move $t$ .

$\frac{64 t - 16 t^{2} + 3 (2 (t \cdot t) + t \cdot 9 - 4 (2 t) - 4 \cdot 9)}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 4.9.1.2.2

Multiply $t$ by $t$ .

$\frac{64 t - 16 t^{2} + 3 (2 t^{2} + t \cdot 9 - 4 (2 t) - 4 \cdot 9)}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 4.9.1.3

Move $9$ to the left of $t$ .

$\frac{64 t - 16 t^{2} + 3 (2 t^{2} + 9 \cdot t - 4 (2 t) - 4 \cdot 9)}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 4.9.1.4

Multiply $2$ by $- 4$ .

$\frac{64 t - 16 t^{2} + 3 (2 t^{2} + 9 t - 8 t - 4 \cdot 9)}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 4.9.1.5

Multiply $- 4$ by $9$ .

$\frac{64 t - 16 t^{2} + 3 (2 t^{2} + 9 t - 8 t - 36)}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 4.9.2

Subtract $8 t$ from $9 t$ .

$\frac{64 t - 16 t^{2} + 3 (2 t^{2} + t - 36)}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 4.10

Apply the distributive property.

$\frac{64 t - 16 t^{2} + 3 (2 t^{2}) + 3 t + 3 \cdot - 36}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 4.11

Simplify.

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Step 4.11.1

Multiply $2$ by $3$ .

$\frac{64 t - 16 t^{2} + 6 t^{2} + 3 t + 3 \cdot - 36}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 4.11.2

Multiply $3$ by $- 36$ .

$\frac{64 t - 16 t^{2} + 6 t^{2} + 3 t - 108}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 5

Simplify by adding terms.

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Step 5.1

Add $64 t$ and $3 t$ .

$\frac{- 16 t^{2} + 6 t^{2} + 67 t - 108}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 5.2

Add $- 16 t^{2}$ and $6 t^{2}$ .

$\frac{- 10 t^{2} + 67 t - 108}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 6

Simplify each term.

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Step 6.1

Factor by grouping.

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Step 6.1.1

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 $a \cdot c = - 10 \cdot - 108 = 1080$ and whose sum is $b = 67$ .

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Step 6.1.1.1

Factor $67$ out of $67 t$ .

$\frac{- 10 t^{2} + 67 (t) - 108}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 6.1.1.2

Rewrite $67$ as $27$ plus $40$

$\frac{- 10 t^{2} + (27 + 40) t - 108}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 6.1.1.3

Apply the distributive property.

$\frac{- 10 t^{2} + 27 t + 40 t - 108}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 6.1.2

Factor out the greatest common factor from each group.

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Step 6.1.2.1

Group the first two terms and the last two terms.

$\frac{(- 10 t^{2} + 27 t) + 40 t - 108}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 6.1.2.2

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

$\frac{t (- 10 t + 27) - 4 (- 10 t + 27)}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 6.1.3

Factor the polynomial by factoring out the greatest common factor, $- 10 t + 27$ .

$\frac{(- 10 t + 27) (t - 4)}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 6.2

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

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Step 6.2.1

Factor $t - 4$ out of $(- 10 t + 27) (t - 4)$ .

$\frac{(t - 4) (- 10 t + 27)}{- 4 {(t - 4)}^{2} (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Factor $t - 4$ out of $- 4 {(t - 4)}^{2} (2 t + 9)$ .

$\frac{(t - 4) (- 10 t + 27)}{(t - 4) (- 4 (t - 4) (2 t + 9))} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 6.2.2.2

Cancel the common factor.

$\frac{(t - 4) (- 10 t + 27)}{(t - 4) (- 4 (t - 4) (2 t + 9))} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 6.2.2.3

Rewrite the expression.

$\frac{- 10 t + 27}{- 4 (t - 4) (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 6.3

Move the negative in front of the fraction.

$- \frac{- 10 t + 27}{(4 (t - 4)) (2 t + 9)} + \frac{- 8 (- 4 {(- t + 4)}^{2})}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 6.4

Cancel the common factor of $- 8$ and $- 4$ .

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Step 6.4.1

Factor $- 4$ out of $- 8 (- 4 {(- t + 4)}^{2})$ .

$- \frac{- 10 t + 27}{4 (t - 4) (2 t + 9)} + \frac{- 4 (2 (- 4 {(- t + 4)}^{2}))}{- 4 {(- t + 4)}^{2} (2 t + 9)}$

Step 6.4.2

Cancel the common factors.

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Step 6.4.2.1

Factor $- 4$ out of $- 4 {(- t + 4)}^{2} (2 t + 9)$ .

$- \frac{- 10 t + 27}{4 (t - 4) (2 t + 9)} + \frac{- 4 (2 (- 4 {(- t + 4)}^{2}))}{- 4 ({(- t + 4)}^{2} (2 t + 9))}$

Step 6.4.2.2

Cancel the common factor.

$- \frac{- 10 t + 27}{4 (t - 4) (2 t + 9)} + \frac{- 4 (2 (- 4 {(- t + 4)}^{2}))}{- 4 ({(- t + 4)}^{2} (2 t + 9))}$

Step 6.4.2.3

Rewrite the expression.

$- \frac{- 10 t + 27}{4 (t - 4) (2 t + 9)} + \frac{2 (- 4 {(- t + 4)}^{2})}{{(- t + 4)}^{2} (2 t + 9)}$

Step 6.5

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

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Step 6.5.1

Cancel the common factor.

$- \frac{- 10 t + 27}{4 (t - 4) (2 t + 9)} + \frac{2 (- 4 {(- t + 4)}^{2})}{{(- t + 4)}^{2} (2 t + 9)}$

Step 6.5.2

Rewrite the expression.

$- \frac{- 10 t + 27}{4 (t - 4) (2 t + 9)} + \frac{2 \cdot (- 4)}{2 t + 9}$

Step 6.6

Multiply $2$ by $- 4$ .

$- \frac{- 10 t + 27}{4 (t - 4) (2 t + 9)} + \frac{- 8}{2 t + 9}$

Step 6.7

Move the negative in front of the fraction.

$- \frac{- 10 t + 27}{4 (t - 4) (2 t + 9)} - \frac{8}{2 t + 9}$

Step 7

To write $- \frac{8}{2 t + 9}$ as a fraction with a common denominator, multiply by $\frac{4 (t - 4)}{4 (t - 4)}$ .

$- \frac{- 10 t + 27}{4 (t - 4) (2 t + 9)} - \frac{8}{2 t + 9} \cdot \frac{4 (t - 4)}{4 (t - 4)}$

Step 8

Write each expression with a common denominator of $4 (t - 4) (2 t + 9)$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.1

Multiply $\frac{8}{2 t + 9}$ by $\frac{4 (t - 4)}{4 (t - 4)}$ .

$- \frac{- 10 t + 27}{4 (t - 4) (2 t + 9)} - \frac{8 (4 (t - 4))}{(2 t + 9) (4 (t - 4))}$

Step 8.2

Reorder the factors of $4 (t - 4) (2 t + 9)$ .

$- \frac{- 10 t + 27}{4 (2 t + 9) (t - 4)} - \frac{8 (4 (t - 4))}{(2 t + 9) (4 (t - 4))}$

Step 8.3

Reorder the factors of $(2 t + 9) (4 (t - 4))$ .

$- \frac{- 10 t + 27}{4 (2 t + 9) (t - 4)} - \frac{8 (4 (t - 4))}{4 (2 t + 9) (t - 4)}$

Step 9

Combine the numerators over the common denominator.

$\frac{- (- 10 t + 27) - 8 (4 (t - 4))}{4 (2 t + 9) (t - 4)}$

Step 10

Simplify the numerator.

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Step 10.1

Factor $- 1$ out of $- (- 10 t + 27) - 8 \cdot 4 (t - 4)$ .

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Step 10.1.1

Factor $- 1$ out of $- 8 \cdot 4 (t - 4)$ .

$\frac{- (- 10 t + 27) - (8 \cdot 4 (t - 4))}{4 (2 t + 9) (t - 4)}$

Step 10.1.2

Factor $- 1$ out of $- (- 10 t + 27) - (8 \cdot 4 (t - 4))$ .

$\frac{- (- 10 t + 27 + 8 \cdot 4 (t - 4))}{4 (2 t + 9) (t - 4)}$

Step 10.2

Multiply $8$ by $4$ .

$\frac{- (- 10 t + 27 + 32 (t - 4))}{4 (2 t + 9) (t - 4)}$

Step 10.3

Apply the distributive property.

$\frac{- (- 10 t + 27 + 32 t + 32 \cdot - 4)}{4 (2 t + 9) (t - 4)}$

Step 10.4

Multiply $32$ by $- 4$ .

$\frac{- (- 10 t + 27 + 32 t - 128)}{4 (2 t + 9) (t - 4)}$

Step 10.5

Add $- 10 t$ and $32 t$ .

$\frac{- (22 t + 27 - 128)}{4 (2 t + 9) (t - 4)}$

Step 10.6

Subtract $128$ from $27$ .

$\frac{- (22 t - 101)}{4 (2 t + 9) (t - 4)}$

Step 11

Move the negative in front of the fraction.

$- \frac{22 t - 101}{4 (2 t + 9) (t - 4)}$