Simplify (a-4)(3+a)(3-2a)

$(a - 4) (3 + a) (3 - 2 a)$

Step 1

Expand $(a - 4) (3 + a)$ using the FOIL Method.

Apply the distributive property.

$(a (3 + a) - 4 (3 + a)) (3 - 2 a)$

Apply the distributive property.

$(a \cdot 3 + a \cdot a - 4 (3 + a)) (3 - 2 a)$

Apply the distributive property.

$(a \cdot 3 + a \cdot a - 4 \cdot 3 - 4 a) (3 - 2 a)$

Step 2

Simplify and combine like terms.

Simplify each term.

Move $3$ to the left of $a$ .

$(3 \cdot a + a \cdot a - 4 \cdot 3 - 4 a) (3 - 2 a)$

Multiply $a$ by $a$ .

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

Multiply $- 4$ by $3$ .

$(3 a + a^{2} - 12 - 4 a) (3 - 2 a)$

Subtract $4 a$ from $3 a$ .

$(- a + a^{2} - 12) (3 - 2 a)$

Step 3

Expand $(- a + a^{2} - 12) (3 - 2 a)$ by multiplying each term in the first expression by each term in the second expression.

$- a \cdot 3 - a (- 2 a) + a^{2} \cdot 3 + a^{2} (- 2 a) - 12 \cdot 3 - 12 (- 2 a)$

Step 4

Simplify terms.

Simplify each term.

Multiply $3$ by $- 1$ .

$- 3 a - a (- 2 a) + a^{2} \cdot 3 + a^{2} (- 2 a) - 12 \cdot 3 - 12 (- 2 a)$

Rewrite using the commutative property of multiplication.

$- 3 a - 1 \cdot - 2 a \cdot a + a^{2} \cdot 3 + a^{2} (- 2 a) - 12 \cdot 3 - 12 (- 2 a)$

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

Move $a$ .

$- 3 a - 1 \cdot - 2 (a \cdot a) + a^{2} \cdot 3 + a^{2} (- 2 a) - 12 \cdot 3 - 12 (- 2 a)$

Multiply $a$ by $a$ .

$- 3 a - 1 \cdot - 2 a^{2} + a^{2} \cdot 3 + a^{2} (- 2 a) - 12 \cdot 3 - 12 (- 2 a)$

Multiply $- 1$ by $- 2$ .

$- 3 a + 2 a^{2} + a^{2} \cdot 3 + a^{2} (- 2 a) - 12 \cdot 3 - 12 (- 2 a)$

Move $3$ to the left of $a^{2}$ .

$- 3 a + 2 a^{2} + 3 \cdot a^{2} + a^{2} (- 2 a) - 12 \cdot 3 - 12 (- 2 a)$

Rewrite using the commutative property of multiplication.

$- 3 a + 2 a^{2} + 3 a^{2} - 2 a^{2} a - 12 \cdot 3 - 12 (- 2 a)$

Multiply $a^{2}$ by $a$ by adding the exponents.

Move $a$ .

$- 3 a + 2 a^{2} + 3 a^{2} - 2 (a \cdot a^{2}) - 12 \cdot 3 - 12 (- 2 a)$

Multiply $a$ by $a^{2}$ .

Raise $a$ to the power of $1$ .

$- 3 a + 2 a^{2} + 3 a^{2} - 2 (a^{1} a^{2}) - 12 \cdot 3 - 12 (- 2 a)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 3 a + 2 a^{2} + 3 a^{2} - 2 a^{1 + 2} - 12 \cdot 3 - 12 (- 2 a)$

Add $1$ and $2$ .

$- 3 a + 2 a^{2} + 3 a^{2} - 2 a^{3} - 12 \cdot 3 - 12 (- 2 a)$

Multiply $- 12$ by $3$ .

$- 3 a + 2 a^{2} + 3 a^{2} - 2 a^{3} - 36 - 12 (- 2 a)$

Multiply $- 2$ by $- 12$ .

$- 3 a + 2 a^{2} + 3 a^{2} - 2 a^{3} - 36 + 24 a$

Simplify by adding terms.

Add $- 3 a$ and $24 a$ .

$2 a^{2} + 3 a^{2} - 2 a^{3} - 36 + 21 a$

Add $2 a^{2}$ and $3 a^{2}$ .

$5 a^{2} - 2 a^{3} - 36 + 21 a$

Simplify the expression.

Move $- 36$ .

$5 a^{2} - 2 a^{3} + 21 a - 36$

Reorder $5 a^{2}$ and $- 2 a^{3}$ .

$- 2 a^{3} + 5 a^{2} + 21 a - 36$

Simplify (a-4)(3+a)(3-2a)

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