Simplify (8a-18)/(3a^2+14a+8)+7/(3a+2)

$\frac{8 a - 18}{3 a^{2} + 14 a + 8} + \frac{7}{3 a + 2}$

Step 1

Simplify each term.

Factor $2$ out of $8 a - 18$ .

Factor $2$ out of $8 a$ .

$\frac{2 (4 a) - 18}{3 a^{2} + 14 a + 8} + \frac{7}{3 a + 2}$

Factor $2$ out of $- 18$ .

$\frac{2 (4 a) + 2 (- 9)}{3 a^{2} + 14 a + 8} + \frac{7}{3 a + 2}$

Factor $2$ out of $2 (4 a) + 2 (- 9)$ .

$\frac{2 (4 a - 9)}{3 a^{2} + 14 a + 8} + \frac{7}{3 a + 2}$

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 $a \cdot c = 3 \cdot 8 = 24$ and whose sum is $b = 14$ .

Factor $14$ out of $14 a$ .

$\frac{2 (4 a - 9)}{3 a^{2} + 14 (a) + 8} + \frac{7}{3 a + 2}$

Rewrite $14$ as $2$ plus $12$

$\frac{2 (4 a - 9)}{3 a^{2} + (2 + 12) a + 8} + \frac{7}{3 a + 2}$

Apply the distributive property.

$\frac{2 (4 a - 9)}{3 a^{2} + 2 a + 12 a + 8} + \frac{7}{3 a + 2}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{2 (4 a - 9)}{(3 a^{2} + 2 a) + 12 a + 8} + \frac{7}{3 a + 2}$

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

$\frac{2 (4 a - 9)}{a (3 a + 2) + 4 (3 a + 2)} + \frac{7}{3 a + 2}$

Factor the polynomial by factoring out the greatest common factor, $3 a + 2$ .

$\frac{2 (4 a - 9)}{(3 a + 2) (a + 4)} + \frac{7}{3 a + 2}$

Step 2

To write $\frac{7}{3 a + 2}$ as a fraction with a common denominator, multiply by $\frac{a + 4}{a + 4}$ .

$\frac{2 (4 a - 9)}{(3 a + 2) (a + 4)} + \frac{7}{3 a + 2} \cdot \frac{a + 4}{a + 4}$

Step 3

Simplify terms.

Multiply $\frac{7}{3 a + 2}$ by $\frac{a + 4}{a + 4}$ .

$\frac{2 (4 a - 9)}{(3 a + 2) (a + 4)} + \frac{7 (a + 4)}{(3 a + 2) (a + 4)}$

Combine the numerators over the common denominator.

$\frac{2 (4 a - 9) + 7 (a + 4)}{(3 a + 2) (a + 4)}$

Step 4

Simplify the numerator.

Apply the distributive property.

$\frac{2 (4 a) + 2 \cdot - 9 + 7 (a + 4)}{(3 a + 2) (a + 4)}$

Multiply $4$ by $2$ .

$\frac{8 a + 2 \cdot - 9 + 7 (a + 4)}{(3 a + 2) (a + 4)}$

Multiply $2$ by $- 9$ .

$\frac{8 a - 18 + 7 (a + 4)}{(3 a + 2) (a + 4)}$

Apply the distributive property.

$\frac{8 a - 18 + 7 a + 7 \cdot 4}{(3 a + 2) (a + 4)}$

Multiply $7$ by $4$ .

$\frac{8 a - 18 + 7 a + 28}{(3 a + 2) (a + 4)}$

Add $8 a$ and $7 a$ .

$\frac{15 a - 18 + 28}{(3 a + 2) (a + 4)}$

Add $- 18$ and $28$ .

$\frac{15 a + 10}{(3 a + 2) (a + 4)}$

Factor $5$ out of $15 a + 10$ .

Factor $5$ out of $15 a$ .

$\frac{5 (3 a) + 10}{(3 a + 2) (a + 4)}$

Factor $5$ out of $10$ .

$\frac{5 (3 a) + 5 (2)}{(3 a + 2) (a + 4)}$

Factor $5$ out of $5 (3 a) + 5 (2)$ .

$\frac{5 (3 a + 2)}{(3 a + 2) (a + 4)}$

Step 5

Cancel the common factor of $3 a + 2$ .

Cancel the common factor.

$\frac{5 (3 a + 2)}{(3 a + 2) (a + 4)}$

Rewrite the expression.

$\frac{5}{a + 4}$

Simplify (8a-18)/(3a^2+14a+8)+7/(3a+2)

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