Find the Product (2y+3)(3y^2+4y+5)

$(2 y + 3) (3 y^{2} + 4 y + 5)$

Step 1

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

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

Step 2

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $y^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

Multiply $2$ by $3$ .

$6 y^{3} + 2 y (4 y) + 2 y \cdot 5 + 3 (3 y^{2}) + 3 (4 y) + 3 \cdot 5$

Rewrite using the commutative property of multiplication.

$6 y^{3} + 2 \cdot 4 y \cdot y + 2 y \cdot 5 + 3 (3 y^{2}) + 3 (4 y) + 3 \cdot 5$

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$6 y^{3} + 2 \cdot 4 (y \cdot y) + 2 y \cdot 5 + 3 (3 y^{2}) + 3 (4 y) + 3 \cdot 5$

Multiply $y$ by $y$ .

$6 y^{3} + 2 \cdot 4 y^{2} + 2 y \cdot 5 + 3 (3 y^{2}) + 3 (4 y) + 3 \cdot 5$

Multiply $2$ by $4$ .

$6 y^{3} + 8 y^{2} + 2 y \cdot 5 + 3 (3 y^{2}) + 3 (4 y) + 3 \cdot 5$

Multiply $5$ by $2$ .

$6 y^{3} + 8 y^{2} + 10 y + 3 (3 y^{2}) + 3 (4 y) + 3 \cdot 5$

Multiply $3$ by $3$ .

$6 y^{3} + 8 y^{2} + 10 y + 9 y^{2} + 3 (4 y) + 3 \cdot 5$

Multiply $4$ by $3$ .

$6 y^{3} + 8 y^{2} + 10 y + 9 y^{2} + 12 y + 3 \cdot 5$

Multiply $3$ by $5$ .

$6 y^{3} + 8 y^{2} + 10 y + 9 y^{2} + 12 y + 15$

Simplify by adding terms.

Add $8 y^{2}$ and $9 y^{2}$ .

$6 y^{3} + 17 y^{2} + 10 y + 12 y + 15$

Add $10 y$ and $12 y$ .

$6 y^{3} + 17 y^{2} + 22 y + 15$