Simplify (x^2-3x-10)/((x+3)/((2x^2-7x-15)/(x^2-9)))

$\frac{x^{2} - 3 x - 10}{\frac{x + 3}{\frac{2 x^{2} - 7 x - 15}{x^{2} - 9}}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$(x^{2} - 3 x - 10) \frac{\frac{2 x^{2} - 7 x - 15}{x^{2} - 9}}{x + 3}$

Step 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 $a \cdot c = 2 \cdot - 15 = - 30$ and whose sum is $b = - 7$ .

Factor $- 7$ out of $- 7 x$ .

$(x^{2} - 3 x - 10) \frac{\frac{2 x^{2} - 7 (x) - 15}{x^{2} - 9}}{x + 3}$

Rewrite $- 7$ as $3$ plus $- 10$

$(x^{2} - 3 x - 10) \frac{\frac{2 x^{2} + (3 - 10) x - 15}{x^{2} - 9}}{x + 3}$

Apply the distributive property.

$(x^{2} - 3 x - 10) \frac{\frac{2 x^{2} + 3 x - 10 x - 15}{x^{2} - 9}}{x + 3}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x^{2} - 3 x - 10) \frac{\frac{(2 x^{2} + 3 x) - 10 x - 15}{x^{2} - 9}}{x + 3}$

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

$(x^{2} - 3 x - 10) \frac{\frac{x (2 x + 3) - 5 (2 x + 3)}{x^{2} - 9}}{x + 3}$

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

$(x^{2} - 3 x - 10) \frac{\frac{(2 x + 3) (x - 5)}{x^{2} - 9}}{x + 3}$

Step 3

Simplify the denominator.

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

$(x^{2} - 3 x - 10) \frac{\frac{(2 x + 3) (x - 5)}{x^{2} - 3^{2}}}{x + 3}$

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 $b = 3$ .

$(x^{2} - 3 x - 10) \frac{\frac{(2 x + 3) (x - 5)}{(x + 3) (x - 3)}}{x + 3}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$(x^{2} - 3 x - 10) (\frac{(2 x + 3) (x - 5)}{(x + 3) (x - 3)} \cdot \frac{1}{x + 3})$

Step 5

Multiply $\frac{(2 x + 3) (x - 5)}{(x + 3) (x - 3)} \cdot \frac{1}{x + 3}$ .

Multiply $\frac{(2 x + 3) (x - 5)}{(x + 3) (x - 3)}$ by $\frac{1}{x + 3}$ .

$(x^{2} - 3 x - 10) \frac{(2 x + 3) (x - 5)}{(x + 3) (x - 3) (x + 3)}$

Raise $x + 3$ to the power of $1$ .

$(x^{2} - 3 x - 10) \frac{(2 x + 3) (x - 5)}{{(x + 3)}^{1} (x + 3) (x - 3)}$

Raise $x + 3$ to the power of $1$ .

$(x^{2} - 3 x - 10) \frac{(2 x + 3) (x - 5)}{{(x + 3)}^{1} {(x + 3)}^{1} (x - 3)}$

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

$(x^{2} - 3 x - 10) \frac{(2 x + 3) (x - 5)}{{(x + 3)}^{1 + 1} (x - 3)}$

Add $1$ and $1$ .

$(x^{2} - 3 x - 10) \frac{(2 x + 3) (x - 5)}{{(x + 3)}^{2} (x - 3)}$

Step 6

Multiply $x^{2} - 3 x - 10$ by $\frac{(2 x + 3) (x - 5)}{{(x + 3)}^{2} (x - 3)}$ .

$\frac{(x^{2} - 3 x - 10) ((2 x + 3) (x - 5))}{{(x + 3)}^{2} (x - 3)}$

Step 7

Simplify the numerator.

Factor $x^{2} - 3 x - 10$ 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 $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Write the factored form using these integers.

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

Combine exponents.

Raise $x - 5$ to the power of $1$ .

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

Raise $x - 5$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Simplify (x^2-3x-10)/((x+3)/((2x^2-7x-15)/(x^2-9)))

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