Simplify (2x^2+13x+20)/(x^2-5x-36)

$\frac{2 x^{2} + 13 x + 20}{x^{2} - 5 x - 36}$

Step 1

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 $a \cdot c = 2 \cdot 20 = 40$ and whose sum is $b = 13$ .

Factor $13$ out of $13 x$ .

$\frac{2 x^{2} + 13 (x) + 20}{x^{2} - 5 x - 36}$

Rewrite $13$ as $5$ plus $8$

$\frac{2 x^{2} + (5 + 8) x + 20}{x^{2} - 5 x - 36}$

Apply the distributive property.

$\frac{2 x^{2} + 5 x + 8 x + 20}{x^{2} - 5 x - 36}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(2 x^{2} + 5 x) + 8 x + 20}{x^{2} - 5 x - 36}$

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

$\frac{x (2 x + 5) + 4 (2 x + 5)}{x^{2} - 5 x - 36}$

Factor the polynomial by factoring out the greatest common factor, $2 x + 5$ .

$\frac{(2 x + 5) (x + 4)}{x^{2} - 5 x - 36}$

Step 2

Factor $x^{2} - 5 x - 36$ 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 $- 36$ and whose sum is $- 5$ .

$- 9, 4$

Write the factored form using these integers.

$\frac{(2 x + 5) (x + 4)}{(x - 9) (x + 4)}$

Step 3

Cancel the common factor of $x + 4$ .

Cancel the common factor.

$\frac{(2 x + 5) (x + 4)}{(x - 9) (x + 4)}$

Rewrite the expression.

$\frac{2 x + 5}{x - 9}$

Simplify (2x^2+13x+20)/(x^2-5x-36)

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