Simplify (x+1)(2x-3)(x-5)(x-1)

$(x + 1) (2 x - 3) (x - 5) (x - 1)$

Step 1

Expand $(x + 1) (2 x - 3)$ using the FOIL Method.

Apply the distributive property.

$(x (2 x - 3) + 1 (2 x - 3)) (x - 5) (x - 1)$

Apply the distributive property.

$(x (2 x) + x \cdot - 3 + 1 (2 x - 3)) (x - 5) (x - 1)$

Apply the distributive property.

$(x (2 x) + x \cdot - 3 + 1 (2 x) + 1 \cdot - 3) (x - 5) (x - 1)$

Step 2

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$(2 x \cdot x + x \cdot - 3 + 1 (2 x) + 1 \cdot - 3) (x - 5) (x - 1)$

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

Move $x$ .

$(2 (x \cdot x) + x \cdot - 3 + 1 (2 x) + 1 \cdot - 3) (x - 5) (x - 1)$

Multiply $x$ by $x$ .

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

Move $- 3$ to the left of $x$ .

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

Multiply $2 x$ by $1$ .

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

Multiply $- 3$ by $1$ .

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

Add $- 3 x$ and $2 x$ .

$(2 x^{2} - x - 3) (x - 5) (x - 1)$

Step 3

Expand $(2 x^{2} - x - 3) (x - 5)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 4

Simplify terms.

Simplify each term.

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Multiply $- 5$ by $2$ .

$(2 x^{3} - 10 x^{2} - x \cdot x - x \cdot - 5 - 3 x - 3 \cdot - 5) (x - 1)$

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

Move $x$ .

$(2 x^{3} - 10 x^{2} - (x \cdot x) - x \cdot - 5 - 3 x - 3 \cdot - 5) (x - 1)$

Multiply $x$ by $x$ .

$(2 x^{3} - 10 x^{2} - x^{2} - x \cdot - 5 - 3 x - 3 \cdot - 5) (x - 1)$

Multiply $- 5$ by $- 1$ .

$(2 x^{3} - 10 x^{2} - x^{2} + 5 x - 3 x - 3 \cdot - 5) (x - 1)$

Multiply $- 3$ by $- 5$ .

$(2 x^{3} - 10 x^{2} - x^{2} + 5 x - 3 x + 15) (x - 1)$

Simplify by adding terms.

Subtract $x^{2}$ from $- 10 x^{2}$ .

$(2 x^{3} - 11 x^{2} + 5 x - 3 x + 15) (x - 1)$

Subtract $3 x$ from $5 x$ .

$(2 x^{3} - 11 x^{2} + 2 x + 15) (x - 1)$

Step 5

Expand $(2 x^{3} - 11 x^{2} + 2 x + 15) (x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$2 x^{3} x + 2 x^{3} \cdot - 1 - 11 x^{2} x - 11 x^{2} \cdot - 1 + 2 x \cdot x + 2 x \cdot - 1 + 15 x + 15 \cdot - 1$

Step 6

Simplify terms.

Simplify each term.

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

Move $x$ .

$2 (x \cdot x^{3}) + 2 x^{3} \cdot - 1 - 11 x^{2} x - 11 x^{2} \cdot - 1 + 2 x \cdot x + 2 x \cdot - 1 + 15 x + 15 \cdot - 1$

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

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

$2 (x^{1} x^{3}) + 2 x^{3} \cdot - 1 - 11 x^{2} x - 11 x^{2} \cdot - 1 + 2 x \cdot x + 2 x \cdot - 1 + 15 x + 15 \cdot - 1$

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

$2 x^{1 + 3} + 2 x^{3} \cdot - 1 - 11 x^{2} x - 11 x^{2} \cdot - 1 + 2 x \cdot x + 2 x \cdot - 1 + 15 x + 15 \cdot - 1$

Add $1$ and $3$ .

$2 x^{4} + 2 x^{3} \cdot - 1 - 11 x^{2} x - 11 x^{2} \cdot - 1 + 2 x \cdot x + 2 x \cdot - 1 + 15 x + 15 \cdot - 1$

Multiply $- 1$ by $2$ .

$2 x^{4} - 2 x^{3} - 11 x^{2} x - 11 x^{2} \cdot - 1 + 2 x \cdot x + 2 x \cdot - 1 + 15 x + 15 \cdot - 1$

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

Move $x$ .

$2 x^{4} - 2 x^{3} - 11 (x \cdot x^{2}) - 11 x^{2} \cdot - 1 + 2 x \cdot x + 2 x \cdot - 1 + 15 x + 15 \cdot - 1$

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

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

$2 x^{4} - 2 x^{3} - 11 (x^{1} x^{2}) - 11 x^{2} \cdot - 1 + 2 x \cdot x + 2 x \cdot - 1 + 15 x + 15 \cdot - 1$

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

$2 x^{4} - 2 x^{3} - 11 x^{1 + 2} - 11 x^{2} \cdot - 1 + 2 x \cdot x + 2 x \cdot - 1 + 15 x + 15 \cdot - 1$

Add $1$ and $2$ .

$2 x^{4} - 2 x^{3} - 11 x^{3} - 11 x^{2} \cdot - 1 + 2 x \cdot x + 2 x \cdot - 1 + 15 x + 15 \cdot - 1$

Multiply $- 1$ by $- 11$ .

$2 x^{4} - 2 x^{3} - 11 x^{3} + 11 x^{2} + 2 x \cdot x + 2 x \cdot - 1 + 15 x + 15 \cdot - 1$

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

Move $x$ .

$2 x^{4} - 2 x^{3} - 11 x^{3} + 11 x^{2} + 2 (x \cdot x) + 2 x \cdot - 1 + 15 x + 15 \cdot - 1$

Multiply $x$ by $x$ .

$2 x^{4} - 2 x^{3} - 11 x^{3} + 11 x^{2} + 2 x^{2} + 2 x \cdot - 1 + 15 x + 15 \cdot - 1$

Multiply $- 1$ by $2$ .

$2 x^{4} - 2 x^{3} - 11 x^{3} + 11 x^{2} + 2 x^{2} - 2 x + 15 x + 15 \cdot - 1$

Multiply $15$ by $- 1$ .

$2 x^{4} - 2 x^{3} - 11 x^{3} + 11 x^{2} + 2 x^{2} - 2 x + 15 x - 15$

Simplify by adding terms.

Subtract $11 x^{3}$ from $- 2 x^{3}$ .

$2 x^{4} - 13 x^{3} + 11 x^{2} + 2 x^{2} - 2 x + 15 x - 15$

Add $11 x^{2}$ and $2 x^{2}$ .

$2 x^{4} - 13 x^{3} + 13 x^{2} - 2 x + 15 x - 15$

Add $- 2 x$ and $15 x$ .

$2 x^{4} - 13 x^{3} + 13 x^{2} + 13 x - 15$

Simplify (x+1)(2x-3)(x-5)(x-1)

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