Precalculus Examples

$3 (5 a + 3 c) (2 a - 3 c) - {(4 a + c)}^{2} - (a + 4 c) (a - 4 c)$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$(3 (5 a) + 3 (3 c)) (2 a - 3 c) - {(4 a + c)}^{2} - (a + 4 c) (a - 4 c)$

Step 1.2

Multiply $5$ by $3$ .

$(15 a + 3 (3 c)) (2 a - 3 c) - {(4 a + c)}^{2} - (a + 4 c) (a - 4 c)$

Step 1.3

Multiply $3$ by $3$ .

$(15 a + 9 c) (2 a - 3 c) - {(4 a + c)}^{2} - (a + 4 c) (a - 4 c)$

Step 1.4

Expand $(15 a + 9 c) (2 a - 3 c)$ using the FOIL Method.

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Step 1.4.1

Apply the distributive property.

$15 a (2 a - 3 c) + 9 c (2 a - 3 c) - {(4 a + c)}^{2} - (a + 4 c) (a - 4 c)$

Step 1.4.2

Apply the distributive property.

$15 a (2 a) + 15 a (- 3 c) + 9 c (2 a - 3 c) - {(4 a + c)}^{2} - (a + 4 c) (a - 4 c)$

Step 1.4.3

Apply the distributive property.

$15 a (2 a) + 15 a (- 3 c) + 9 c (2 a) + 9 c (- 3 c) - {(4 a + c)}^{2} - (a + 4 c) (a - 4 c)$

Step 1.5

Simplify and combine like terms.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Rewrite using the commutative property of multiplication.

$15 \cdot 2 a \cdot a + 15 a (- 3 c) + 9 c (2 a) + 9 c (- 3 c) - {(4 a + c)}^{2} - (a + 4 c) (a - 4 c)$

Step 1.5.1.2

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

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Step 1.5.1.2.1

Move $a$ .

$15 \cdot 2 (a \cdot a) + 15 a (- 3 c) + 9 c (2 a) + 9 c (- 3 c) - {(4 a + c)}^{2} - (a + 4 c) (a - 4 c)$

Step 1.5.1.2.2

Multiply $a$ by $a$ .

$15 \cdot 2 a^{2} + 15 a (- 3 c) + 9 c (2 a) + 9 c (- 3 c) - {(4 a + c)}^{2} - (a + 4 c) (a - 4 c)$

Step 1.5.1.3

Multiply $15$ by $2$ .

$30 a^{2} + 15 a (- 3 c) + 9 c (2 a) + 9 c (- 3 c) - {(4 a + c)}^{2} - (a + 4 c) (a - 4 c)$

Step 1.5.1.4

Rewrite using the commutative property of multiplication.

$30 a^{2} + 15 \cdot - 3 a c + 9 c (2 a) + 9 c (- 3 c) - {(4 a + c)}^{2} - (a + 4 c) (a - 4 c)$

Step 1.5.1.5

Multiply $15$ by $- 3$ .

$30 a^{2} - 45 a c + 9 c (2 a) + 9 c (- 3 c) - {(4 a + c)}^{2} - (a + 4 c) (a - 4 c)$

Step 1.5.1.6

Rewrite using the commutative property of multiplication.

$30 a^{2} - 45 a c + 9 \cdot 2 c a + 9 c (- 3 c) - {(4 a + c)}^{2} - (a + 4 c) (a - 4 c)$

Step 1.5.1.7

Multiply $9$ by $2$ .

$30 a^{2} - 45 a c + 18 c a + 9 c (- 3 c) - {(4 a + c)}^{2} - (a + 4 c) (a - 4 c)$

Step 1.5.1.8

Rewrite using the commutative property of multiplication.

$30 a^{2} - 45 a c + 18 c a + 9 \cdot - 3 c \cdot c - {(4 a + c)}^{2} - (a + 4 c) (a - 4 c)$

Step 1.5.1.9

Multiply $c$ by $c$ by adding the exponents.

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Step 1.5.1.9.1

Move $c$ .

$30 a^{2} - 45 a c + 18 c a + 9 \cdot - 3 (c \cdot c) - {(4 a + c)}^{2} - (a + 4 c) (a - 4 c)$

Step 1.5.1.9.2

Multiply $c$ by $c$ .

$30 a^{2} - 45 a c + 18 c a + 9 \cdot - 3 c^{2} - {(4 a + c)}^{2} - (a + 4 c) (a - 4 c)$

Step 1.5.1.10

Multiply $9$ by $- 3$ .

$30 a^{2} - 45 a c + 18 c a - 27 c^{2} - {(4 a + c)}^{2} - (a + 4 c) (a - 4 c)$

Step 1.5.2

Add $- 45 a c$ and $18 c a$ .

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Step 1.5.2.1

Move $c$ .

$30 a^{2} - 45 a c + 18 a c - 27 c^{2} - {(4 a + c)}^{2} - (a + 4 c) (a - 4 c)$

Step 1.5.2.2

Add $- 45 a c$ and $18 a c$ .

$30 a^{2} - 27 a c - 27 c^{2} - {(4 a + c)}^{2} - (a + 4 c) (a - 4 c)$

Step 1.6

Rewrite ${(4 a + c)}^{2}$ as $(4 a + c) (4 a + c)$ .

$30 a^{2} - 27 a c - 27 c^{2} - ((4 a + c) (4 a + c)) - (a + 4 c) (a - 4 c)$

Step 1.7

Expand $(4 a + c) (4 a + c)$ using the FOIL Method.

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Step 1.7.1

Apply the distributive property.

$30 a^{2} - 27 a c - 27 c^{2} - (4 a (4 a + c) + c (4 a + c)) - (a + 4 c) (a - 4 c)$

Step 1.7.2

Apply the distributive property.

$30 a^{2} - 27 a c - 27 c^{2} - (4 a (4 a) + 4 a c + c (4 a + c)) - (a + 4 c) (a - 4 c)$

Step 1.7.3

Apply the distributive property.

$30 a^{2} - 27 a c - 27 c^{2} - (4 a (4 a) + 4 a c + c (4 a) + c \cdot c) - (a + 4 c) (a - 4 c)$

Step 1.8

Simplify and combine like terms.

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Rewrite using the commutative property of multiplication.

$30 a^{2} - 27 a c - 27 c^{2} - (4 \cdot 4 a \cdot a + 4 a c + c (4 a) + c \cdot c) - (a + 4 c) (a - 4 c)$

Step 1.8.1.2

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

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Step 1.8.1.2.1

Move $a$ .

$30 a^{2} - 27 a c - 27 c^{2} - (4 \cdot 4 (a \cdot a) + 4 a c + c (4 a) + c \cdot c) - (a + 4 c) (a - 4 c)$

Step 1.8.1.2.2

Multiply $a$ by $a$ .

$30 a^{2} - 27 a c - 27 c^{2} - (4 \cdot 4 a^{2} + 4 a c + c (4 a) + c \cdot c) - (a + 4 c) (a - 4 c)$

Step 1.8.1.3

Multiply $4$ by $4$ .

$30 a^{2} - 27 a c - 27 c^{2} - (16 a^{2} + 4 a c + c (4 a) + c \cdot c) - (a + 4 c) (a - 4 c)$

Step 1.8.1.4

Rewrite using the commutative property of multiplication.

$30 a^{2} - 27 a c - 27 c^{2} - (16 a^{2} + 4 a c + 4 c a + c \cdot c) - (a + 4 c) (a - 4 c)$

Step 1.8.1.5

Multiply $c$ by $c$ .

$30 a^{2} - 27 a c - 27 c^{2} - (16 a^{2} + 4 a c + 4 c a + c^{2}) - (a + 4 c) (a - 4 c)$

Step 1.8.2

Add $4 a c$ and $4 c a$ .

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Step 1.8.2.1

Move $c$ .

$30 a^{2} - 27 a c - 27 c^{2} - (16 a^{2} + 4 a c + 4 a c + c^{2}) - (a + 4 c) (a - 4 c)$

Step 1.8.2.2

Add $4 a c$ and $4 a c$ .

$30 a^{2} - 27 a c - 27 c^{2} - (16 a^{2} + 8 a c + c^{2}) - (a + 4 c) (a - 4 c)$

Step 1.9

Apply the distributive property.

$30 a^{2} - 27 a c - 27 c^{2} - (16 a^{2}) - (8 a c) - c^{2} - (a + 4 c) (a - 4 c)$

Step 1.10

Simplify.

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Step 1.10.1

Multiply $16$ by $- 1$ .

$30 a^{2} - 27 a c - 27 c^{2} - 16 a^{2} - (8 a c) - c^{2} - (a + 4 c) (a - 4 c)$

Step 1.10.2

Multiply $8$ by $- 1$ .

$30 a^{2} - 27 a c - 27 c^{2} - 16 a^{2} - 8 (a c) - c^{2} - (a + 4 c) (a - 4 c)$

Step 1.11

Apply the distributive property.

$30 a^{2} - 27 a c - 27 c^{2} - 16 a^{2} - 8 a c - c^{2} + (- a - (4 c)) (a - 4 c)$

Step 1.12

Multiply $4$ by $- 1$ .

$30 a^{2} - 27 a c - 27 c^{2} - 16 a^{2} - 8 a c - c^{2} + (- a - 4 c) (a - 4 c)$

Step 1.13

Expand $(- a - 4 c) (a - 4 c)$ using the FOIL Method.

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Step 1.13.1

Apply the distributive property.

$30 a^{2} - 27 a c - 27 c^{2} - 16 a^{2} - 8 a c - c^{2} - a (a - 4 c) - 4 c (a - 4 c)$

Step 1.13.2

Apply the distributive property.

$30 a^{2} - 27 a c - 27 c^{2} - 16 a^{2} - 8 a c - c^{2} - a \cdot a - a (- 4 c) - 4 c (a - 4 c)$

Step 1.13.3

Apply the distributive property.

$30 a^{2} - 27 a c - 27 c^{2} - 16 a^{2} - 8 a c - c^{2} - a \cdot a - a (- 4 c) - 4 c a - 4 c (- 4 c)$

Step 1.14

Simplify and combine like terms.

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Step 1.14.1

Simplify each term.

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Step 1.14.1.1

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

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Step 1.14.1.1.1

Move $a$ .

$30 a^{2} - 27 a c - 27 c^{2} - 16 a^{2} - 8 a c - c^{2} - (a \cdot a) - a (- 4 c) - 4 c a - 4 c (- 4 c)$

Step 1.14.1.1.2

Multiply $a$ by $a$ .

$30 a^{2} - 27 a c - 27 c^{2} - 16 a^{2} - 8 a c - c^{2} - a^{2} - a (- 4 c) - 4 c a - 4 c (- 4 c)$

Step 1.14.1.2

Rewrite using the commutative property of multiplication.

$30 a^{2} - 27 a c - 27 c^{2} - 16 a^{2} - 8 a c - c^{2} - a^{2} - 1 \cdot - 4 a c - 4 c a - 4 c (- 4 c)$

Step 1.14.1.3

Multiply $- 1$ by $- 4$ .

$30 a^{2} - 27 a c - 27 c^{2} - 16 a^{2} - 8 a c - c^{2} - a^{2} + 4 a c - 4 c a - 4 c (- 4 c)$

Step 1.14.1.4

Rewrite using the commutative property of multiplication.

$30 a^{2} - 27 a c - 27 c^{2} - 16 a^{2} - 8 a c - c^{2} - a^{2} + 4 a c - 4 c a - 4 \cdot - 4 c \cdot c$

Step 1.14.1.5

Multiply $c$ by $c$ by adding the exponents.

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Step 1.14.1.5.1

Move $c$ .

$30 a^{2} - 27 a c - 27 c^{2} - 16 a^{2} - 8 a c - c^{2} - a^{2} + 4 a c - 4 c a - 4 \cdot - 4 (c \cdot c)$

Step 1.14.1.5.2

Multiply $c$ by $c$ .

$30 a^{2} - 27 a c - 27 c^{2} - 16 a^{2} - 8 a c - c^{2} - a^{2} + 4 a c - 4 c a - 4 \cdot - 4 c^{2}$

Step 1.14.1.6

Multiply $- 4$ by $- 4$ .

$30 a^{2} - 27 a c - 27 c^{2} - 16 a^{2} - 8 a c - c^{2} - a^{2} + 4 a c - 4 c a + 16 c^{2}$

Step 1.14.2

Subtract $4 c a$ from $4 a c$ .

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Step 1.14.2.1

Move $c$ .

$30 a^{2} - 27 a c - 27 c^{2} - 16 a^{2} - 8 a c - c^{2} - a^{2} + 4 a c - 4 a c + 16 c^{2}$

Step 1.14.2.2

Subtract $4 a c$ from $4 a c$ .

$30 a^{2} - 27 a c - 27 c^{2} - 16 a^{2} - 8 a c - c^{2} - a^{2} + 0 + 16 c^{2}$

Step 1.14.3

Add $- a^{2}$ and $0$ .

$30 a^{2} - 27 a c - 27 c^{2} - 16 a^{2} - 8 a c - c^{2} - a^{2} + 16 c^{2}$

Step 2

Simplify by adding terms.

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Step 2.1

Subtract $16 a^{2}$ from $30 a^{2}$ .

$14 a^{2} - 27 a c - 27 c^{2} - 8 a c - c^{2} - a^{2} + 16 c^{2}$

Step 2.2

Subtract $a^{2}$ from $14 a^{2}$ .

$13 a^{2} - 27 a c - 27 c^{2} - 8 a c - c^{2} + 16 c^{2}$

Step 2.3

Subtract $8 a c$ from $- 27 a c$ .

$13 a^{2} - 27 c^{2} - 35 a c - c^{2} + 16 c^{2}$

Step 2.4

Subtract $c^{2}$ from $- 27 c^{2}$ .

$13 a^{2} - 28 c^{2} - 35 a c + 16 c^{2}$

Step 2.5

Add $- 28 c^{2}$ and $16 c^{2}$ .

$13 a^{2} - 12 c^{2} - 35 a c$