Factor x^2-a^2+2xy+y^2+2ab-b^2

$x^{2} - a^{2} + 2 x y + y^{2} + 2 a b - b^{2}$

Step 1

Regroup terms.

$x^{2} + 2 x y + y^{2} - a^{2} + 2 a b - b^{2}$

Step 2

Factor using the perfect square rule.

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

$2 x y = 2 \cdot x \cdot y$

Rewrite the polynomial.

$x^{2} + 2 \cdot x \cdot y + y^{2} - a^{2} + 2 a b - b^{2}$

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

${(x + y)}^{2} - a^{2} + 2 a b - b^{2}$

Step 3

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 $a \cdot c = - 1 \cdot - 1 = 1$ and whose sum is $b = 2$ .

Reorder terms.

${(x + y)}^{2} - a^{2} - b^{2} + 2 a b$

Reorder $- b^{2}$ and $2 a b$ .

${(x + y)}^{2} - a^{2} + 2 a b - b^{2}$

Factor $2$ out of $2 a b$ .

${(x + y)}^{2} - a^{2} + 2 (a b) - b^{2}$

Rewrite $2$ as $1$ plus $1$

${(x + y)}^{2} - a^{2} + (1 + 1) (a b) - b^{2}$

Apply the distributive property.

${(x + y)}^{2} - a^{2} + 1 (a b) + 1 (a b) - b^{2}$

Multiply $a b$ by $1$ .

${(x + y)}^{2} - a^{2} + a b + 1 (a b) - b^{2}$

Multiply $a b$ by $1$ .

${(x + y)}^{2} - a^{2} + a b + a b - b^{2}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

${(x + y)}^{2} + (- a^{2} + a b) + a b - b^{2}$

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

${(x + y)}^{2} + a (- a + b) - b (- a + b)$

Factor the polynomial by factoring out the greatest common factor, $- a + b$ .

${(x + y)}^{2} + (- a + b) (a - b)$

Step 4

Rewrite $(a - b) (a - b)$ as ${(a - b)}^{2}$ .

${(x + y)}^{2} - {(a - b)}^{2}$

Step 5

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

$(x + y + a - b) (x + y - (a - b))$

Step 6

Simplify.

Apply the distributive property.

$(x + y + a - b) (x + y - a - - b)$

Multiply $- - b$ .

Multiply $- 1$ by $- 1$ .

$(x + y + a - b) (x + y - a + 1 b)$

Multiply $b$ by $1$ .

$(x + y + a - b) (x + y - a + b)$

Factor x^2-a^2+2xy+y^2+2ab-b^2

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