Factor by Grouping (a+b)(a-b)+a(a+b)

$(a + b) (a - b) + a (a + b)$

Step 1

Simplify each term.

Expand $(a + b) (a - b)$ using the FOIL Method.

Apply the distributive property.

$a (a - b) + b (a - b) + a (a + b)$

Apply the distributive property.

$a \cdot a + a (- b) + b (a - b) + a (a + b)$

Apply the distributive property.

$a \cdot a + a (- b) + b a + b (- b) + a (a + b)$

Combine the opposite terms in $a \cdot a + a (- b) + b a + b (- b)$ .

Reorder the factors in the terms $a (- b)$ and $b a$ .

$a \cdot a - a b + a b + b (- b) + a (a + b)$

Add $- a b$ and $a b$ .

$a \cdot a + 0 + b (- b) + a (a + b)$

Add $a \cdot a$ and $0$ .

$a \cdot a + b (- b) + a (a + b)$

Simplify each term.

Multiply $a$ by $a$ .

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

Rewrite using the commutative property of multiplication.

$a^{2} - b \cdot b + a (a + b)$

Multiply $b$ by $b$ by adding the exponents.

Move $b$ .

$a^{2} - (b \cdot b) + a (a + b)$

Multiply $b$ by $b$ .

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

Apply the distributive property.

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

Multiply $a$ by $a$ .

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

Step 2

Add $a^{2}$ and $a^{2}$ .

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

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

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

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

Factor $1$ out of $a b$ .

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

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

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

Apply the distributive property.

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

Remove unnecessary parentheses.

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

Remove unnecessary parentheses.

$2 a^{2} - 1 a b + 2 a b - b^{2}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

$a (2 a - 1 b) + b (2 a - b)$

Factor the polynomial by factoring out the greatest common factor, $2 a - 1 b$ .

$(2 a - 1 b) (a + b)$

Step 4

Rewrite $- 1 b$ as $- b$ .

$(2 a - b) (a + b)$

Factor by Grouping (a+b)(a-b)+a(a+b)

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