Simplify (a-1)(2a+3)(4a-5)

$(a - 1) (2 a + 3) (4 a - 5)$

Step 1

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

Apply the distributive property.

$(a (2 a + 3) - 1 (2 a + 3)) (4 a - 5)$

Apply the distributive property.

$(a (2 a) + a \cdot 3 - 1 (2 a + 3)) (4 a - 5)$

Apply the distributive property.

$(a (2 a) + a \cdot 3 - 1 (2 a) - 1 \cdot 3) (4 a - 5)$

Step 2

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$(2 a \cdot a + a \cdot 3 - 1 (2 a) - 1 \cdot 3) (4 a - 5)$

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

Move $a$ .

$(2 (a \cdot a) + a \cdot 3 - 1 (2 a) - 1 \cdot 3) (4 a - 5)$

Multiply $a$ by $a$ .

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

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

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

Multiply $2$ by $- 1$ .

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

Multiply $- 1$ by $3$ .

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

Subtract $2 a$ from $3 a$ .

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

Step 3

Expand $(2 a^{2} + a - 3) (4 a - 5)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 4

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $a$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Multiply $2$ by $4$ .

$8 a^{3} + 2 a^{2} \cdot - 5 + a (4 a) + a \cdot - 5 - 3 (4 a) - 3 \cdot - 5$

Multiply $- 5$ by $2$ .

$8 a^{3} - 10 a^{2} + a (4 a) + a \cdot - 5 - 3 (4 a) - 3 \cdot - 5$

Rewrite using the commutative property of multiplication.

$8 a^{3} - 10 a^{2} + 4 a \cdot a + a \cdot - 5 - 3 (4 a) - 3 \cdot - 5$

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

Move $a$ .

$8 a^{3} - 10 a^{2} + 4 (a \cdot a) + a \cdot - 5 - 3 (4 a) - 3 \cdot - 5$

Multiply $a$ by $a$ .

$8 a^{3} - 10 a^{2} + 4 a^{2} + a \cdot - 5 - 3 (4 a) - 3 \cdot - 5$

Move $- 5$ to the left of $a$ .

$8 a^{3} - 10 a^{2} + 4 a^{2} - 5 \cdot a - 3 (4 a) - 3 \cdot - 5$

Multiply $4$ by $- 3$ .

$8 a^{3} - 10 a^{2} + 4 a^{2} - 5 a - 12 a - 3 \cdot - 5$

Multiply $- 3$ by $- 5$ .

$8 a^{3} - 10 a^{2} + 4 a^{2} - 5 a - 12 a + 15$

Simplify by adding terms.

Add $- 10 a^{2}$ and $4 a^{2}$ .

$8 a^{3} - 6 a^{2} - 5 a - 12 a + 15$

Subtract $12 a$ from $- 5 a$ .

$8 a^{3} - 6 a^{2} - 17 a + 15$

Simplify (a-1)(2a+3)(4a-5)

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