Expand the Trigonometric Expression (d^2+3)(d^2+2d+1)

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

Step 1

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

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

Step 2

Simplify terms.

Simplify each term.

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

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

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

Add $2$ and $2$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $d$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

Multiply $2$ by $3$ .

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

Multiply $3$ by $1$ .

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

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

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