Simplify (-3x^3+12x^2+34x-58)/(x^2-9)+(5x^2+7x+5)/(-x^2+9)

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58}{x^{2} - 9} + \frac{5 x^{2} + 7 x + 5}{- x^{2} + 9}$

Step 1

Simplify terms.

Simplify each term.

Simplify the denominator.

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

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58}{x^{2} - 3^{2}} + \frac{5 x^{2} + 7 x + 5}{- x^{2} + 9}$

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

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58}{(x + 3) (x - 3)} + \frac{5 x^{2} + 7 x + 5}{- x^{2} + 9}$

Simplify the denominator.

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

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58}{(x + 3) (x - 3)} + \frac{5 x^{2} + 7 x + 5}{- x^{2} + 3^{2}}$

Reorder $- x^{2}$ and $3^{2}$ .

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58}{(x + 3) (x - 3)} + \frac{5 x^{2} + 7 x + 5}{3^{2} - x^{2}}$

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 $b = x$ .

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58}{(x + 3) (x - 3)} + \frac{5 x^{2} + 7 x + 5}{(3 + x) (3 - x)}$

Simplify terms.

Reorder terms.

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58}{(x + 3) (x - 3)} + \frac{5 x^{2} + 7 x + 5}{(x + 3) (3 - x)}$

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

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58}{(x + 3) (x - 3)} + \frac{5 x^{2} + 7 x + 5}{(x + 3) (- 1 (- 3) - x)}$

Factor $- 1$ out of $- x$ .

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58}{(x + 3) (x - 3)} + \frac{5 x^{2} + 7 x + 5}{(x + 3) (- 1 (- 3) - (x))}$

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

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58}{(x + 3) (x - 3)} + \frac{5 x^{2} + 7 x + 5}{(x + 3) (- 1 (- 3 + x))}$

Simplify the expression.

Move a negative from the denominator of $\frac{5 x^{2} + 7 x + 5}{(x + 3) (- 1 (- 3 + x))}$ to the numerator.

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58}{(x + 3) (x - 3)} + \frac{- (5 x^{2} + 7 x + 5)}{(x + 3) (- 3 + x)}$

Reorder terms.

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58}{(x + 3) (x - 3)} + \frac{- (5 x^{2} + 7 x + 5)}{(x + 3) (x - 3)}$

Combine the numerators over the common denominator.

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58 - (5 x^{2} + 7 x + 5)}{(x + 3) (x - 3)}$

Step 2

Simplify the numerator.

Apply the distributive property.

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58 - (5 x^{2}) - (7 x) - 1 \cdot 5}{(x + 3) (x - 3)}$

Simplify.

Multiply $5$ by $- 1$ .

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58 - 5 x^{2} - (7 x) - 1 \cdot 5}{(x + 3) (x - 3)}$

Multiply $7$ by $- 1$ .

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58 - 5 x^{2} - 7 x - 1 \cdot 5}{(x + 3) (x - 3)}$

Multiply $- 1$ by $5$ .

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58 - 5 x^{2} - 7 x - 5}{(x + 3) (x - 3)}$

Subtract $5 x^{2}$ from $12 x^{2}$ .

$\frac{- 3 x^{3} + 7 x^{2} + 34 x - 58 - 7 x - 5}{(x + 3) (x - 3)}$

Subtract $7 x$ from $34 x$ .

$\frac{- 3 x^{3} + 7 x^{2} + 27 x - 58 - 5}{(x + 3) (x - 3)}$

Subtract $5$ from $- 58$ .

$\frac{- 3 x^{3} + 7 x^{2} + 27 x - 63}{(x + 3) (x - 3)}$

Rewrite $- 3 x^{3} + 7 x^{2} + 27 x - 63$ in a factored form.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(- 3 x^{3} + 7 x^{2}) + 27 x - 63}{(x + 3) (x - 3)}$

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

$\frac{x^{2} (- 3 x + 7) - 9 (- 3 x + 7)}{(x + 3) (x - 3)}$

Factor the polynomial by factoring out the greatest common factor, $- 3 x + 7$ .

$\frac{(- 3 x + 7) (x^{2} - 9)}{(x + 3) (x - 3)}$

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

$\frac{(- 3 x + 7) (x^{2} - 3^{2})}{(x + 3) (x - 3)}$

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

$\frac{(- 3 x + 7) (x + 3) (x - 3)}{(x + 3) (x - 3)}$

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x + 3$ .

Cancel the common factor.

$\frac{(- 3 x + 7) (x + 3) (x - 3)}{(x + 3) (x - 3)}$

Rewrite the expression.

$\frac{(- 3 x + 7) (x - 3)}{x - 3}$

Cancel the common factor of $x - 3$ .

Cancel the common factor.

$\frac{(- 3 x + 7) (x - 3)}{x - 3}$

Divide $- 3 x + 7$ by $1$ .

$- 3 x + 7$

Simplify (-3x^3+12x^2+34x-58)/(x^2-9)+(5x^2+7x+5)/(-x^2+9)

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