Find the Product (x^2+3x+1)(x^2+x+2)

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

Step 1

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

$x^{2} x^{2} + x^{2} x + x^{2} \cdot 2 + 3 x \cdot x^{2} + 3 x \cdot x + 3 x \cdot 2 + 1 x^{2} + 1 x + 1 \cdot 2$

Step 2

Simplify terms.

Simplify each term.

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

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

$x^{2 + 2} + x^{2} x + x^{2} \cdot 2 + 3 x \cdot x^{2} + 3 x \cdot x + 3 x \cdot 2 + 1 x^{2} + 1 x + 1 \cdot 2$

Add $2$ and $2$ .

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

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

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

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

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

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

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

Add $2$ and $1$ .

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

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

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

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

Move $x^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $2$ by $3$ .

$x^{4} + x^{3} + 2 x^{2} + 3 x^{3} + 3 x^{2} + 6 x + 1 x^{2} + 1 x + 1 \cdot 2$

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

$x^{4} + x^{3} + 2 x^{2} + 3 x^{3} + 3 x^{2} + 6 x + x^{2} + 1 x + 1 \cdot 2$

Multiply $x$ by $1$ .

$x^{4} + x^{3} + 2 x^{2} + 3 x^{3} + 3 x^{2} + 6 x + x^{2} + x + 1 \cdot 2$

Multiply $2$ by $1$ .

$x^{4} + x^{3} + 2 x^{2} + 3 x^{3} + 3 x^{2} + 6 x + x^{2} + x + 2$

Simplify by adding terms.

Add $x^{3}$ and $3 x^{3}$ .

$x^{4} + 4 x^{3} + 2 x^{2} + 3 x^{2} + 6 x + x^{2} + x + 2$

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

$x^{4} + 4 x^{3} + 5 x^{2} + 6 x + x^{2} + x + 2$

Add $5 x^{2}$ and $x^{2}$ .

$x^{4} + 4 x^{3} + 6 x^{2} + 6 x + x + 2$

Add $6 x$ and $x$ .

$x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$

Find the Product (x^2+3x+1)(x^2+x+2)

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