Simplify (a+15)/(9-a^2)-(a+5)/(a+3)+(a-5)/(3-a)

$\frac{a + 15}{9 - a^{2}} - \frac{a + 5}{a + 3} + \frac{a - 5}{3 - a}$

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

$\frac{a + 15}{3^{2} - a^{2}} - \frac{a + 5}{a + 3} + \frac{a - 5}{3 - a}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = a$ .

$\frac{a + 15}{(3 + a) (3 - a)} - \frac{a + 5}{a + 3} + \frac{a - 5}{3 - a}$

Reorder terms.

$\frac{a + 15}{(a + 3) (3 - a)} - \frac{a + 5}{a + 3} + \frac{a - 5}{3 - a}$

To write $- \frac{a + 5}{a + 3}$ as a fraction with a common denominator, multiply by $\frac{3 - a}{3 - a}$ .

$\frac{a + 15}{(a + 3) (3 - a)} - \frac{a + 5}{a + 3} \cdot \frac{3 - a}{3 - a} + \frac{a - 5}{3 - a}$

Simplify terms.

Multiply $\frac{a + 5}{a + 3}$ by $\frac{3 - a}{3 - a}$ .

$\frac{a + 15}{(a + 3) (3 - a)} - \frac{(a + 5) (3 - a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Combine the numerators over the common denominator.

$\frac{a + 15 - (a + 5) (3 - a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Simplify each term.

Simplify the numerator.

Apply the distributive property.

$\frac{a + 15 + (- a - 1 \cdot 5) (3 - a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Multiply $- 1$ by $5$ .

$\frac{a + 15 + (- a - 5) (3 - a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Expand $(- a - 5) (3 - a)$ using the FOIL Method.

Apply the distributive property.

$\frac{a + 15 - a (3 - a) - 5 (3 - a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Apply the distributive property.

$\frac{a + 15 - a \cdot 3 - a (- a) - 5 (3 - a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Apply the distributive property.

$\frac{a + 15 - a \cdot 3 - a (- a) - 5 \cdot 3 - 5 (- a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $- 1$ .

$\frac{a + 15 - 3 a - a (- a) - 5 \cdot 3 - 5 (- a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Rewrite using the commutative property of multiplication.

$\frac{a + 15 - 3 a - 1 \cdot - 1 a \cdot a - 5 \cdot 3 - 5 (- a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Multiply $a$ by $a$ by adding the exponents.

Move $a$ .

$\frac{a + 15 - 3 a - 1 \cdot - 1 (a \cdot a) - 5 \cdot 3 - 5 (- a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Multiply $a$ by $a$ .

$\frac{a + 15 - 3 a - 1 \cdot - 1 a^{2} - 5 \cdot 3 - 5 (- a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Multiply $- 1$ by $- 1$ .

$\frac{a + 15 - 3 a + 1 a^{2} - 5 \cdot 3 - 5 (- a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Multiply $a^{2}$ by $1$ .

$\frac{a + 15 - 3 a + a^{2} - 5 \cdot 3 - 5 (- a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Multiply $- 5$ by $3$ .

$\frac{a + 15 - 3 a + a^{2} - 15 - 5 (- a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Multiply $- 1$ by $- 5$ .

$\frac{a + 15 - 3 a + a^{2} - 15 + 5 a}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Add $- 3 a$ and $5 a$ .

$\frac{a + 15 + 2 a + a^{2} - 15}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Add $a$ and $2 a$ .

$\frac{3 a + 15 + a^{2} - 15}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Subtract $15$ from $15$ .

$\frac{3 a + a^{2} + 0}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Add $3 a + a^{2}$ and $0$ .

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

Factor $a$ out of $3 a + a^{2}$ .

Factor $a$ out of $3 a$ .

$\frac{a \cdot 3 + a^{2}}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Factor $a$ out of $a^{2}$ .

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

Factor $a$ out of $a \cdot 3 + a \cdot a$ .

$\frac{a (3 + a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

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

Reorder terms.

$\frac{a (a + 3)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Cancel the common factor.

$\frac{a (a + 3)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Rewrite the expression.

$\frac{a}{3 - a} + \frac{a - 5}{3 - a}$

Simplify terms.

Combine the numerators over the common denominator.

$\frac{a + a - 5}{- a + 3}$

Add $a$ and $a$ .

$\frac{2 a - 5}{- a + 3}$

Factor $- 1$ out of $- a$ .

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

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{2 a - 5}{- (a) - 1 (- 3)}$

Factor $- 1$ out of $- (a) - 1 (- 3)$ .

$\frac{2 a - 5}{- (a - 3)}$

Rewrite negatives.

Rewrite $- (a - 3)$ as $- 1 (a - 3)$ .

$\frac{2 a - 5}{- 1 (a - 3)}$

Move the negative in front of the fraction.

$- \frac{2 a - 5}{a - 3}$

Simplify (a+15)/(9-a^2)-(a+5)/(a+3)+(a-5)/(3-a)

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