Simplify (5x^2+20xy+20y^2)/(x^2-xy-6y^2)

$\frac{5 x^{2} + 20 x y + 20 y^{2}}{x^{2} - x y - 6 y^{2}}$

Simplify the numerator.

Factor $5$ out of $5 x^{2} + 20 x y + 20 y^{2}$ .

Factor $5$ out of $5 x^{2}$ .

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

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

$\frac{5 (x^{2}) + 5 (4 x y) + 20 y^{2}}{x^{2} - x y - 6 y^{2}}$

Factor $5$ out of $20 y^{2}$ .

$\frac{5 (x^{2}) + 5 (4 x y) + 5 (4 y^{2})}{x^{2} - x y - 6 y^{2}}$

Factor $5$ out of $5 (x^{2}) + 5 (4 x y)$ .

$\frac{5 (x^{2} + 4 x y) + 5 (4 y^{2})}{x^{2} - x y - 6 y^{2}}$

Factor $5$ out of $5 (x^{2} + 4 x y) + 5 (4 y^{2})$ .

$\frac{5 (x^{2} + 4 x y + 4 y^{2})}{x^{2} - x y - 6 y^{2}}$

Factor using the perfect square rule.

Rewrite $4 y^{2}$ as ${(2 y)}^{2}$ .

$\frac{5 (x^{2} + 4 x y + {(2 y)}^{2})}{x^{2} - x y - 6 y^{2}}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x y = 2 \cdot x \cdot (2 y)$

Rewrite the polynomial.

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

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 2 y$ .

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

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 $a \cdot c = 1 \cdot - 6 = - 6$ and whose sum is $b = - 1$ .

Reorder terms.

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

Reorder $- 6 y^{2}$ and $- x y$ .

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

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

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

Rewrite $- 1$ as $2$ plus $- 3$

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

Apply the distributive property.

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

Move parentheses.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

Cancel the common factor of ${(x + 2 y)}^{2}$ and $x + 2 y$ .

Factor $x + 2 y$ out of $5 {(x + 2 y)}^{2}$ .

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

Cancel the common factors.

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (5x^2+20xy+20y^2)/(x^2-xy-6y^2)

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