Factor (a+x)^2-2(a+x)(x+y)+(x+y)^2

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

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

Reorder terms.

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

Reorder ${(x + y)}^{2}$ and $- 2 (a + x) (x + y)$ .

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

Factor $- 2$ out of $- 2 (a + x) (x + y)$ .

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

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

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

Apply the distributive property.

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

Remove unnecessary parentheses.

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

Remove unnecessary parentheses.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

Factor the polynomial by factoring out the greatest common factor, $a + x - 1 (x + y)$ .

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

Apply the distributive property.

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

Rewrite $- 1 x$ as $- x$ .

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

Rewrite $- 1 y$ as $- y$ .

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

Subtract $x$ from $x$ .

$(a + 0 - y) (a + x - (x + y))$

Add $a$ and $0$ .

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

Apply the distributive property.

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

Subtract $x$ from $x$ .

$(a - y) (a + 0 - y)$

Add $a$ and $0$ .

$(a - y) (a - y)$

Factor (a+x)^2-2(a+x)(x+y)+(x+y)^2

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