Simplify (x-1)(x-4)(x+2)(3x+2)

$(x - 1) (x - 4) (x + 2) (3 x + 2)$

Step 1

Expand $(x - 1) (x - 4)$ using the FOIL Method.

Apply the distributive property.

$(x (x - 4) - 1 (x - 4)) (x + 2) (3 x + 2)$

Apply the distributive property.

$(x \cdot x + x \cdot - 4 - 1 (x - 4)) (x + 2) (3 x + 2)$

Apply the distributive property.

$(x \cdot x + x \cdot - 4 - 1 x - 1 \cdot - 4) (x + 2) (3 x + 2)$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$(x^{2} + x \cdot - 4 - 1 x - 1 \cdot - 4) (x + 2) (3 x + 2)$

Move $- 4$ to the left of $x$ .

$(x^{2} - 4 \cdot x - 1 x - 1 \cdot - 4) (x + 2) (3 x + 2)$

Rewrite $- 1 x$ as $- x$ .

$(x^{2} - 4 x - x - 1 \cdot - 4) (x + 2) (3 x + 2)$

Multiply $- 1$ by $- 4$ .

$(x^{2} - 4 x - x + 4) (x + 2) (3 x + 2)$

Subtract $x$ from $- 4 x$ .

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

Step 3

Expand $(x^{2} - 5 x + 4) (x + 2)$ by multiplying each term in the first expression by each term in the second expression.

$(x^{2} x + x^{2} \cdot 2 - 5 x \cdot x - 5 x \cdot 2 + 4 x + 4 \cdot 2) (3 x + 2)$

Step 4

Simplify terms.

Simplify each term.

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

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

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

$(x^{2} x^{1} + x^{2} \cdot 2 - 5 x \cdot x - 5 x \cdot 2 + 4 x + 4 \cdot 2) (3 x + 2)$

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

$(x^{2 + 1} + x^{2} \cdot 2 - 5 x \cdot x - 5 x \cdot 2 + 4 x + 4 \cdot 2) (3 x + 2)$

Add $2$ and $1$ .

$(x^{3} + x^{2} \cdot 2 - 5 x \cdot x - 5 x \cdot 2 + 4 x + 4 \cdot 2) (3 x + 2)$

Move $2$ to the left of $x^{2}$ .

$(x^{3} + 2 \cdot x^{2} - 5 x \cdot x - 5 x \cdot 2 + 4 x + 4 \cdot 2) (3 x + 2)$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$(x^{3} + 2 x^{2} - 5 (x \cdot x) - 5 x \cdot 2 + 4 x + 4 \cdot 2) (3 x + 2)$

Multiply $x$ by $x$ .

$(x^{3} + 2 x^{2} - 5 x^{2} - 5 x \cdot 2 + 4 x + 4 \cdot 2) (3 x + 2)$

Multiply $2$ by $- 5$ .

$(x^{3} + 2 x^{2} - 5 x^{2} - 10 x + 4 x + 4 \cdot 2) (3 x + 2)$

Multiply $4$ by $2$ .

$(x^{3} + 2 x^{2} - 5 x^{2} - 10 x + 4 x + 8) (3 x + 2)$

Simplify by adding terms.

Subtract $5 x^{2}$ from $2 x^{2}$ .

$(x^{3} - 3 x^{2} - 10 x + 4 x + 8) (3 x + 2)$

Add $- 10 x$ and $4 x$ .

$(x^{3} - 3 x^{2} - 6 x + 8) (3 x + 2)$

Step 5

Expand $(x^{3} - 3 x^{2} - 6 x + 8) (3 x + 2)$ by multiplying each term in the first expression by each term in the second expression.

$x^{3} (3 x) + x^{3} \cdot 2 - 3 x^{2} (3 x) - 3 x^{2} \cdot 2 - 6 x (3 x) - 6 x \cdot 2 + 8 (3 x) + 8 \cdot 2$

Step 6

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$3 x^{3} x + x^{3} \cdot 2 - 3 x^{2} (3 x) - 3 x^{2} \cdot 2 - 6 x (3 x) - 6 x \cdot 2 + 8 (3 x) + 8 \cdot 2$

Multiply $x^{3}$ by $x$ by adding the exponents.

Move $x$ .

$3 (x \cdot x^{3}) + x^{3} \cdot 2 - 3 x^{2} (3 x) - 3 x^{2} \cdot 2 - 6 x (3 x) - 6 x \cdot 2 + 8 (3 x) + 8 \cdot 2$

Multiply $x$ by $x^{3}$ .

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

$3 (x^{1} x^{3}) + x^{3} \cdot 2 - 3 x^{2} (3 x) - 3 x^{2} \cdot 2 - 6 x (3 x) - 6 x \cdot 2 + 8 (3 x) + 8 \cdot 2$

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

$3 x^{1 + 3} + x^{3} \cdot 2 - 3 x^{2} (3 x) - 3 x^{2} \cdot 2 - 6 x (3 x) - 6 x \cdot 2 + 8 (3 x) + 8 \cdot 2$

Add $1$ and $3$ .

$3 x^{4} + x^{3} \cdot 2 - 3 x^{2} (3 x) - 3 x^{2} \cdot 2 - 6 x (3 x) - 6 x \cdot 2 + 8 (3 x) + 8 \cdot 2$

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

$3 x^{4} + 2 \cdot x^{3} - 3 x^{2} (3 x) - 3 x^{2} \cdot 2 - 6 x (3 x) - 6 x \cdot 2 + 8 (3 x) + 8 \cdot 2$

Rewrite using the commutative property of multiplication.

$3 x^{4} + 2 x^{3} - 3 \cdot 3 x^{2} x - 3 x^{2} \cdot 2 - 6 x (3 x) - 6 x \cdot 2 + 8 (3 x) + 8 \cdot 2$

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

Move $x$ .

$3 x^{4} + 2 x^{3} - 3 \cdot 3 (x \cdot x^{2}) - 3 x^{2} \cdot 2 - 6 x (3 x) - 6 x \cdot 2 + 8 (3 x) + 8 \cdot 2$

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

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

$3 x^{4} + 2 x^{3} - 3 \cdot 3 (x^{1} x^{2}) - 3 x^{2} \cdot 2 - 6 x (3 x) - 6 x \cdot 2 + 8 (3 x) + 8 \cdot 2$

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

$3 x^{4} + 2 x^{3} - 3 \cdot 3 x^{1 + 2} - 3 x^{2} \cdot 2 - 6 x (3 x) - 6 x \cdot 2 + 8 (3 x) + 8 \cdot 2$

Add $1$ and $2$ .

$3 x^{4} + 2 x^{3} - 3 \cdot 3 x^{3} - 3 x^{2} \cdot 2 - 6 x (3 x) - 6 x \cdot 2 + 8 (3 x) + 8 \cdot 2$

Multiply $- 3$ by $3$ .

$3 x^{4} + 2 x^{3} - 9 x^{3} - 3 x^{2} \cdot 2 - 6 x (3 x) - 6 x \cdot 2 + 8 (3 x) + 8 \cdot 2$

Multiply $2$ by $- 3$ .

$3 x^{4} + 2 x^{3} - 9 x^{3} - 6 x^{2} - 6 x (3 x) - 6 x \cdot 2 + 8 (3 x) + 8 \cdot 2$

Rewrite using the commutative property of multiplication.

$3 x^{4} + 2 x^{3} - 9 x^{3} - 6 x^{2} - 6 \cdot 3 x \cdot x - 6 x \cdot 2 + 8 (3 x) + 8 \cdot 2$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$3 x^{4} + 2 x^{3} - 9 x^{3} - 6 x^{2} - 6 \cdot 3 (x \cdot x) - 6 x \cdot 2 + 8 (3 x) + 8 \cdot 2$

Multiply $x$ by $x$ .

$3 x^{4} + 2 x^{3} - 9 x^{3} - 6 x^{2} - 6 \cdot 3 x^{2} - 6 x \cdot 2 + 8 (3 x) + 8 \cdot 2$

Multiply $- 6$ by $3$ .

$3 x^{4} + 2 x^{3} - 9 x^{3} - 6 x^{2} - 18 x^{2} - 6 x \cdot 2 + 8 (3 x) + 8 \cdot 2$

Multiply $2$ by $- 6$ .

$3 x^{4} + 2 x^{3} - 9 x^{3} - 6 x^{2} - 18 x^{2} - 12 x + 8 (3 x) + 8 \cdot 2$

Multiply $3$ by $8$ .

$3 x^{4} + 2 x^{3} - 9 x^{3} - 6 x^{2} - 18 x^{2} - 12 x + 24 x + 8 \cdot 2$

Multiply $8$ by $2$ .

$3 x^{4} + 2 x^{3} - 9 x^{3} - 6 x^{2} - 18 x^{2} - 12 x + 24 x + 16$

Simplify by adding terms.

Subtract $9 x^{3}$ from $2 x^{3}$ .

$3 x^{4} - 7 x^{3} - 6 x^{2} - 18 x^{2} - 12 x + 24 x + 16$

Subtract $18 x^{2}$ from $- 6 x^{2}$ .

$3 x^{4} - 7 x^{3} - 24 x^{2} - 12 x + 24 x + 16$

Add $- 12 x$ and $24 x$ .

$3 x^{4} - 7 x^{3} - 24 x^{2} + 12 x + 16$

Simplify (x-1)(x-4)(x+2)(3x+2)

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