Expand the Trigonometric Expression (15a-4n-6)^2

${(15 a - 4 n - 6)}^{2}$

Step 1

Rewrite ${(15 a - 4 n - 6)}^{2}$ as $(15 a - 4 n - 6) (15 a - 4 n - 6)$ .

$(15 a - 4 n - 6) (15 a - 4 n - 6)$

Step 2

Expand $(15 a - 4 n - 6) (15 a - 4 n - 6)$ by multiplying each term in the first expression by each term in the second expression.

$15 a (15 a) + 15 a (- 4 n) + 15 a \cdot - 6 - 4 n (15 a) - 4 n (- 4 n) - 4 n \cdot - 6 - 6 (15 a) - 6 (- 4 n) - 6 \cdot - 6$

Step 3

Simplify each term.

Rewrite using the commutative property of multiplication.

$15 \cdot 15 a \cdot a + 15 a (- 4 n) + 15 a \cdot - 6 - 4 n (15 a) - 4 n (- 4 n) - 4 n \cdot - 6 - 6 (15 a) - 6 (- 4 n) - 6 \cdot - 6$

Multiply $a$ by $a$ by adding the exponents.

Move $a$ .

$15 \cdot 15 (a \cdot a) + 15 a (- 4 n) + 15 a \cdot - 6 - 4 n (15 a) - 4 n (- 4 n) - 4 n \cdot - 6 - 6 (15 a) - 6 (- 4 n) - 6 \cdot - 6$

Multiply $a$ by $a$ .

$15 \cdot 15 a^{2} + 15 a (- 4 n) + 15 a \cdot - 6 - 4 n (15 a) - 4 n (- 4 n) - 4 n \cdot - 6 - 6 (15 a) - 6 (- 4 n) - 6 \cdot - 6$

Multiply $15$ by $15$ .

$225 a^{2} + 15 a (- 4 n) + 15 a \cdot - 6 - 4 n (15 a) - 4 n (- 4 n) - 4 n \cdot - 6 - 6 (15 a) - 6 (- 4 n) - 6 \cdot - 6$

Rewrite using the commutative property of multiplication.

$225 a^{2} + 15 \cdot - 4 a n + 15 a \cdot - 6 - 4 n (15 a) - 4 n (- 4 n) - 4 n \cdot - 6 - 6 (15 a) - 6 (- 4 n) - 6 \cdot - 6$

Multiply $15$ by $- 4$ .

$225 a^{2} - 60 a n + 15 a \cdot - 6 - 4 n (15 a) - 4 n (- 4 n) - 4 n \cdot - 6 - 6 (15 a) - 6 (- 4 n) - 6 \cdot - 6$

Multiply $- 6$ by $15$ .

$225 a^{2} - 60 a n - 90 a - 4 n (15 a) - 4 n (- 4 n) - 4 n \cdot - 6 - 6 (15 a) - 6 (- 4 n) - 6 \cdot - 6$

Rewrite using the commutative property of multiplication.

$225 a^{2} - 60 a n - 90 a - 4 \cdot 15 n a - 4 n (- 4 n) - 4 n \cdot - 6 - 6 (15 a) - 6 (- 4 n) - 6 \cdot - 6$

Multiply $- 4$ by $15$ .

$225 a^{2} - 60 a n - 90 a - 60 n a - 4 n (- 4 n) - 4 n \cdot - 6 - 6 (15 a) - 6 (- 4 n) - 6 \cdot - 6$

Rewrite using the commutative property of multiplication.

$225 a^{2} - 60 a n - 90 a - 60 n a - 4 \cdot - 4 n \cdot n - 4 n \cdot - 6 - 6 (15 a) - 6 (- 4 n) - 6 \cdot - 6$

Multiply $n$ by $n$ by adding the exponents.

Move $n$ .

$225 a^{2} - 60 a n - 90 a - 60 n a - 4 \cdot - 4 (n \cdot n) - 4 n \cdot - 6 - 6 (15 a) - 6 (- 4 n) - 6 \cdot - 6$

Multiply $n$ by $n$ .

$225 a^{2} - 60 a n - 90 a - 60 n a - 4 \cdot - 4 n^{2} - 4 n \cdot - 6 - 6 (15 a) - 6 (- 4 n) - 6 \cdot - 6$

Multiply $- 4$ by $- 4$ .

$225 a^{2} - 60 a n - 90 a - 60 n a + 16 n^{2} - 4 n \cdot - 6 - 6 (15 a) - 6 (- 4 n) - 6 \cdot - 6$

Multiply $- 6$ by $- 4$ .

$225 a^{2} - 60 a n - 90 a - 60 n a + 16 n^{2} + 24 n - 6 (15 a) - 6 (- 4 n) - 6 \cdot - 6$

Multiply $15$ by $- 6$ .

$225 a^{2} - 60 a n - 90 a - 60 n a + 16 n^{2} + 24 n - 90 a - 6 (- 4 n) - 6 \cdot - 6$

Multiply $- 4$ by $- 6$ .

$225 a^{2} - 60 a n - 90 a - 60 n a + 16 n^{2} + 24 n - 90 a + 24 n - 6 \cdot - 6$

Multiply $- 6$ by $- 6$ .

$225 a^{2} - 60 a n - 90 a - 60 n a + 16 n^{2} + 24 n - 90 a + 24 n + 36$

Step 4

Subtract $60 n a$ from $- 60 a n$ .

Move $n$ .

$225 a^{2} - 90 a - 60 a n - 60 a n + 16 n^{2} + 24 n - 90 a + 24 n + 36$

Subtract $60 a n$ from $- 60 a n$ .

$225 a^{2} - 90 a - 120 a n + 16 n^{2} + 24 n - 90 a + 24 n + 36$

Step 5

Subtract $90 a$ from $- 90 a$ .

$225 a^{2} - 180 a - 120 a n + 16 n^{2} + 24 n + 24 n + 36$

Step 6

Add $24 n$ and $24 n$ .

$225 a^{2} - 180 a - 120 a n + 16 n^{2} + 48 n + 36$

Expand the Trigonometric Expression (15a-4n-6)^2

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