Write the Fraction in Simplest Form (3x^2)/(x^2-x-30)-(3x+5)/(x-6)

$\frac{3 x^{2}}{x^{2} - x - 30} - \frac{3 x + 5}{x - 6}$

Factor $x^{2} - x - 30$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 30$ and whose sum is $- 1$ .

$- 6, 5$

Write the factored form using these integers.

$\frac{3 x^{2}}{(x - 6) (x + 5)} - \frac{3 x + 5}{x - 6}$

To write $- \frac{3 x + 5}{x - 6}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{3 x^{2}}{(x - 6) (x + 5)} - \frac{3 x + 5}{x - 6} \cdot \frac{x + 5}{x + 5}$

Multiply $\frac{3 x + 5}{x - 6}$ by $\frac{x + 5}{x + 5}$ .

$\frac{3 x^{2}}{(x - 6) (x + 5)} - \frac{(3 x + 5) (x + 5)}{(x - 6) (x + 5)}$

Combine the numerators over the common denominator.

$\frac{3 x^{2} - (3 x + 5) (x + 5)}{(x - 6) (x + 5)}$

Simplify the numerator.

Apply the distributive property.

$\frac{3 x^{2} + (- (3 x) - 1 \cdot 5) (x + 5)}{(x - 6) (x + 5)}$

Multiply $3$ by $- 1$ .

$\frac{3 x^{2} + (- 3 x - 1 \cdot 5) (x + 5)}{(x - 6) (x + 5)}$

Multiply $- 1$ by $5$ .

$\frac{3 x^{2} + (- 3 x - 5) (x + 5)}{(x - 6) (x + 5)}$

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

Apply the distributive property.

$\frac{3 x^{2} - 3 x (x + 5) - 5 (x + 5)}{(x - 6) (x + 5)}$

Apply the distributive property.

$\frac{3 x^{2} - 3 x \cdot x - 3 x \cdot 5 - 5 (x + 5)}{(x - 6) (x + 5)}$

Apply the distributive property.

$\frac{3 x^{2} - 3 x \cdot x - 3 x \cdot 5 - 5 x - 5 \cdot 5}{(x - 6) (x + 5)}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{3 x^{2} - 3 (x \cdot x) - 3 x \cdot 5 - 5 x - 5 \cdot 5}{(x - 6) (x + 5)}$

Multiply $x$ by $x$ .

$\frac{3 x^{2} - 3 x^{2} - 3 x \cdot 5 - 5 x - 5 \cdot 5}{(x - 6) (x + 5)}$

Multiply $5$ by $- 3$ .

$\frac{3 x^{2} - 3 x^{2} - 15 x - 5 x - 5 \cdot 5}{(x - 6) (x + 5)}$

Multiply $- 5$ by $5$ .

$\frac{3 x^{2} - 3 x^{2} - 15 x - 5 x - 25}{(x - 6) (x + 5)}$

Subtract $5 x$ from $- 15 x$ .

$\frac{3 x^{2} - 3 x^{2} - 20 x - 25}{(x - 6) (x + 5)}$

Subtract $3 x^{2}$ from $3 x^{2}$ .

$\frac{0 - 20 x - 25}{(x - 6) (x + 5)}$

Subtract $20 x$ from $0$ .

$\frac{- 20 x - 25}{(x - 6) (x + 5)}$

Factor $5$ out of $- 20 x - 25$ .

Factor $5$ out of $- 20 x$ .

$\frac{5 (- 4 x) - 25}{(x - 6) (x + 5)}$

Factor $5$ out of $- 25$ .

$\frac{5 (- 4 x) + 5 (- 5)}{(x - 6) (x + 5)}$

Factor $5$ out of $5 (- 4 x) + 5 (- 5)$ .

$\frac{5 (- 4 x - 5)}{(x - 6) (x + 5)}$

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

$\frac{5 (- (4 x) - 5)}{(x - 6) (x + 5)}$

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{5 (- (4 x) - 1 (5))}{(x - 6) (x + 5)}$

Factor $- 1$ out of $- (4 x) - 1 (5)$ .

$\frac{5 (- (4 x + 5))}{(x - 6) (x + 5)}$

Rewrite $- (4 x + 5)$ as $- 1 (4 x + 5)$ .

$\frac{5 (- 1 (4 x + 5))}{(x - 6) (x + 5)}$

Move the negative in front of the fraction.

$- \frac{5 (4 x + 5)}{(x - 6) (x + 5)}$

Write the Fraction in Simplest Form (3x^2)/(x^2-x-30)-(3x+5)/(x-6)

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