Factor (x+y)^2-2z(x+y)-15z^2

${(x + y)}^{2} - 2 z (x + y) - 15 z^{2}$

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 $a \cdot c = - 15 \cdot 1 = - 15$ and whose sum is $b = - 2$ .

Reorder terms.

$- 15 z^{2} + {(x + y)}^{2} - 2 z (x + y)$

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

$- 15 z^{2} - 2 z (x + y) + {(x + y)}^{2}$

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

$- 15 z^{2} - 2 (z (x + y)) + {(x + y)}^{2}$

Rewrite $- 2$ as $3$ plus $- 5$

$- 15 z^{2} + (3 - 5) (z (x + y)) + {(x + y)}^{2}$

Apply the distributive property.

$- 15 z^{2} + 3 (z (x + y)) - 5 (z (x + y)) + {(x + y)}^{2}$

Remove unnecessary parentheses.

$- 15 z^{2} + 3 z (x + y) - 5 (z (x + y)) + {(x + y)}^{2}$

Remove unnecessary parentheses.

$- 15 z^{2} + 3 z (x + y) - 5 z (x + y) + {(x + y)}^{2}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(- 15 z^{2} + 3 z (x + y)) - 5 z (x + y) + {(x + y)}^{2}$

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

$3 z (- 5 z + x + y) + (x + y) (- 5 z + x + y)$

Factor the polynomial by factoring out the greatest common factor, $- 5 z + x + y$ .

$(- 5 z + x + y) (3 z + x + y)$

Step 2

Factor $- 1$ out of $- 5 z + x + y$ .

Move $- 5 z$ .

$(x + y - 5 z) (3 z + x + y)$

Factor $- 1$ out of $x$ .

$(- 1 (- x) + y - 5 z) (3 z + x + y)$

Factor $- 1$ out of $y$ .

$(- 1 (- x) - 1 (- y) - 5 z) (3 z + x + y)$

Factor $- 1$ out of $- 5 z$ .

$(- 1 (- x) - 1 (- y) - (5 z)) (3 z + x + y)$

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

$(- 1 (- x - y) - (5 z)) (3 z + x + y)$

Factor $- 1$ out of $- 1 (- x - y) - (5 z)$ .

$- 1 (- x - y + 5 z) (3 z + x + y)$

Rewrite $- 1 (- x - y + 5 z)$ as $- (- x - y + 5 z)$ .

$- (- x - y + 5 z) (3 z + x + y)$

Step 3

Factor.

Factor $- 1$ out of $- x - y + 5 z$ .

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

$- (- (x) - y + 5 z) (3 z + x + y)$

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

$- (- (x) - (y) + 5 z) (3 z + x + y)$

Factor $- 1$ out of $5 z$ .

$- (- (x) - (y) - (- 5 z)) (3 z + x + y)$

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

$- (- (x + y) - (- 5 z)) (3 z + x + y)$

Factor $- 1$ out of $- (x + y) - (- 5 z)$ .

$- - (x + y - 5 z) (3 z + x + y)$

Remove unnecessary parentheses.

$- - 1 (x + y - 5 z) (3 z + x + y)$

Step 4

Combine exponents.

Factor out negative.

$- ((- (x + y - 5 z)) (3 z + x + y))$

Multiply $- 1$ by $- 1$ .

$1 ((x + y - 5 z) (3 z + x + y))$

Multiply $x + y - 5 z$ by $1$ .

$(x + y - 5 z) (3 z + x + y)$

Factor (x+y)^2-2z(x+y)-15z^2

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