Simplify (y^2+3y+7)(8y^2+y+1)

$(y^{2} + 3 y + 7) (8 y^{2} + y + 1)$

Step 1

Expand $(y^{2} + 3 y + 7) (8 y^{2} + y + 1)$ by multiplying each term in the first expression by each term in the second expression.

$y^{2} (8 y^{2}) + y^{2} y + y^{2} \cdot 1 + 3 y (8 y^{2}) + 3 y \cdot y + 3 y \cdot 1 + 7 (8 y^{2}) + 7 y + 7 \cdot 1$

Step 2

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$8 y^{2} y^{2} + y^{2} y + y^{2} \cdot 1 + 3 y (8 y^{2}) + 3 y \cdot y + 3 y \cdot 1 + 7 (8 y^{2}) + 7 y + 7 \cdot 1$

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

Move $y^{2}$ .

$8 (y^{2} y^{2}) + y^{2} y + y^{2} \cdot 1 + 3 y (8 y^{2}) + 3 y \cdot y + 3 y \cdot 1 + 7 (8 y^{2}) + 7 y + 7 \cdot 1$

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

$8 y^{2 + 2} + y^{2} y + y^{2} \cdot 1 + 3 y (8 y^{2}) + 3 y \cdot y + 3 y \cdot 1 + 7 (8 y^{2}) + 7 y + 7 \cdot 1$

Add $2$ and $2$ .

$8 y^{4} + y^{2} y + y^{2} \cdot 1 + 3 y (8 y^{2}) + 3 y \cdot y + 3 y \cdot 1 + 7 (8 y^{2}) + 7 y + 7 \cdot 1$

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

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

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

$8 y^{4} + y^{2} y^{1} + y^{2} \cdot 1 + 3 y (8 y^{2}) + 3 y \cdot y + 3 y \cdot 1 + 7 (8 y^{2}) + 7 y + 7 \cdot 1$

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

$8 y^{4} + y^{2 + 1} + y^{2} \cdot 1 + 3 y (8 y^{2}) + 3 y \cdot y + 3 y \cdot 1 + 7 (8 y^{2}) + 7 y + 7 \cdot 1$

Add $2$ and $1$ .

$8 y^{4} + y^{3} + y^{2} \cdot 1 + 3 y (8 y^{2}) + 3 y \cdot y + 3 y \cdot 1 + 7 (8 y^{2}) + 7 y + 7 \cdot 1$

Multiply $y^{2}$ by $1$ .

$8 y^{4} + y^{3} + y^{2} + 3 y (8 y^{2}) + 3 y \cdot y + 3 y \cdot 1 + 7 (8 y^{2}) + 7 y + 7 \cdot 1$

Rewrite using the commutative property of multiplication.

$8 y^{4} + y^{3} + y^{2} + 3 \cdot 8 y \cdot y^{2} + 3 y \cdot y + 3 y \cdot 1 + 7 (8 y^{2}) + 7 y + 7 \cdot 1$

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

Move $y^{2}$ .

$8 y^{4} + y^{3} + y^{2} + 3 \cdot 8 (y^{2} y) + 3 y \cdot y + 3 y \cdot 1 + 7 (8 y^{2}) + 7 y + 7 \cdot 1$

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

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

$8 y^{4} + y^{3} + y^{2} + 3 \cdot 8 (y^{2} y^{1}) + 3 y \cdot y + 3 y \cdot 1 + 7 (8 y^{2}) + 7 y + 7 \cdot 1$

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

$8 y^{4} + y^{3} + y^{2} + 3 \cdot 8 y^{2 + 1} + 3 y \cdot y + 3 y \cdot 1 + 7 (8 y^{2}) + 7 y + 7 \cdot 1$

Add $2$ and $1$ .

$8 y^{4} + y^{3} + y^{2} + 3 \cdot 8 y^{3} + 3 y \cdot y + 3 y \cdot 1 + 7 (8 y^{2}) + 7 y + 7 \cdot 1$

Multiply $3$ by $8$ .

$8 y^{4} + y^{3} + y^{2} + 24 y^{3} + 3 y \cdot y + 3 y \cdot 1 + 7 (8 y^{2}) + 7 y + 7 \cdot 1$

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

Move $y$ .

$8 y^{4} + y^{3} + y^{2} + 24 y^{3} + 3 (y \cdot y) + 3 y \cdot 1 + 7 (8 y^{2}) + 7 y + 7 \cdot 1$

Multiply $y$ by $y$ .

$8 y^{4} + y^{3} + y^{2} + 24 y^{3} + 3 y^{2} + 3 y \cdot 1 + 7 (8 y^{2}) + 7 y + 7 \cdot 1$

Multiply $3$ by $1$ .

$8 y^{4} + y^{3} + y^{2} + 24 y^{3} + 3 y^{2} + 3 y + 7 (8 y^{2}) + 7 y + 7 \cdot 1$

Multiply $8$ by $7$ .

$8 y^{4} + y^{3} + y^{2} + 24 y^{3} + 3 y^{2} + 3 y + 56 y^{2} + 7 y + 7 \cdot 1$

Multiply $7$ by $1$ .

$8 y^{4} + y^{3} + y^{2} + 24 y^{3} + 3 y^{2} + 3 y + 56 y^{2} + 7 y + 7$

Simplify by adding terms.

Add $y^{3}$ and $24 y^{3}$ .

$8 y^{4} + 25 y^{3} + y^{2} + 3 y^{2} + 3 y + 56 y^{2} + 7 y + 7$

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

$8 y^{4} + 25 y^{3} + 4 y^{2} + 3 y + 56 y^{2} + 7 y + 7$

Add $4 y^{2}$ and $56 y^{2}$ .

$8 y^{4} + 25 y^{3} + 60 y^{2} + 3 y + 7 y + 7$

Add $3 y$ and $7 y$ .

$8 y^{4} + 25 y^{3} + 60 y^{2} + 10 y + 7$

Simplify (y^2+3y+7)(8y^2+y+1)

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