Simplify (n+4)^2(3n^2-7n+1)

${(n + 4)}^{2} (3 n^{2} - 7 n + 1)$

Rewrite ${(n + 4)}^{2}$ as $(n + 4) (n + 4)$ .

$(n + 4) (n + 4) (3 n^{2} - 7 n + 1)$

Expand $(n + 4) (n + 4)$ using the FOIL Method.

Apply the distributive property.

$(n (n + 4) + 4 (n + 4)) (3 n^{2} - 7 n + 1)$

Apply the distributive property.

$(n \cdot n + n \cdot 4 + 4 (n + 4)) (3 n^{2} - 7 n + 1)$

Apply the distributive property.

$(n \cdot n + n \cdot 4 + 4 n + 4 \cdot 4) (3 n^{2} - 7 n + 1)$

Simplify and combine like terms.

Simplify each term.

Multiply $n$ by $n$ .

$(n^{2} + n \cdot 4 + 4 n + 4 \cdot 4) (3 n^{2} - 7 n + 1)$

Move $4$ to the left of $n$ .

$(n^{2} + 4 \cdot n + 4 n + 4 \cdot 4) (3 n^{2} - 7 n + 1)$

Multiply $4$ by $4$ .

$(n^{2} + 4 n + 4 n + 16) (3 n^{2} - 7 n + 1)$

Add $4 n$ and $4 n$ .

$(n^{2} + 8 n + 16) (3 n^{2} - 7 n + 1)$

Expand $(n^{2} + 8 n + 16) (3 n^{2} - 7 n + 1)$ by multiplying each term in the first expression by each term in the second expression.

$n^{2} (3 n^{2}) + n^{2} (- 7 n) + n^{2} \cdot 1 + 8 n (3 n^{2}) + 8 n (- 7 n) + 8 n \cdot 1 + 16 (3 n^{2}) + 16 (- 7 n) + 16 \cdot 1$

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$3 n^{2} n^{2} + n^{2} (- 7 n) + n^{2} \cdot 1 + 8 n (3 n^{2}) + 8 n (- 7 n) + 8 n \cdot 1 + 16 (3 n^{2}) + 16 (- 7 n) + 16 \cdot 1$

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

Move $n^{2}$ .

$3 (n^{2} n^{2}) + n^{2} (- 7 n) + n^{2} \cdot 1 + 8 n (3 n^{2}) + 8 n (- 7 n) + 8 n \cdot 1 + 16 (3 n^{2}) + 16 (- 7 n) + 16 \cdot 1$

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

$3 n^{2 + 2} + n^{2} (- 7 n) + n^{2} \cdot 1 + 8 n (3 n^{2}) + 8 n (- 7 n) + 8 n \cdot 1 + 16 (3 n^{2}) + 16 (- 7 n) + 16 \cdot 1$

Add $2$ and $2$ .

$3 n^{4} + n^{2} (- 7 n) + n^{2} \cdot 1 + 8 n (3 n^{2}) + 8 n (- 7 n) + 8 n \cdot 1 + 16 (3 n^{2}) + 16 (- 7 n) + 16 \cdot 1$

Rewrite using the commutative property of multiplication.

$3 n^{4} - 7 n^{2} n + n^{2} \cdot 1 + 8 n (3 n^{2}) + 8 n (- 7 n) + 8 n \cdot 1 + 16 (3 n^{2}) + 16 (- 7 n) + 16 \cdot 1$

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

Move $n$ .

$3 n^{4} - 7 (n \cdot n^{2}) + n^{2} \cdot 1 + 8 n (3 n^{2}) + 8 n (- 7 n) + 8 n \cdot 1 + 16 (3 n^{2}) + 16 (- 7 n) + 16 \cdot 1$

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

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

$3 n^{4} - 7 (n^{1} n^{2}) + n^{2} \cdot 1 + 8 n (3 n^{2}) + 8 n (- 7 n) + 8 n \cdot 1 + 16 (3 n^{2}) + 16 (- 7 n) + 16 \cdot 1$

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

$3 n^{4} - 7 n^{1 + 2} + n^{2} \cdot 1 + 8 n (3 n^{2}) + 8 n (- 7 n) + 8 n \cdot 1 + 16 (3 n^{2}) + 16 (- 7 n) + 16 \cdot 1$

Add $1$ and $2$ .

$3 n^{4} - 7 n^{3} + n^{2} \cdot 1 + 8 n (3 n^{2}) + 8 n (- 7 n) + 8 n \cdot 1 + 16 (3 n^{2}) + 16 (- 7 n) + 16 \cdot 1$

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

$3 n^{4} - 7 n^{3} + n^{2} + 8 n (3 n^{2}) + 8 n (- 7 n) + 8 n \cdot 1 + 16 (3 n^{2}) + 16 (- 7 n) + 16 \cdot 1$

Rewrite using the commutative property of multiplication.

$3 n^{4} - 7 n^{3} + n^{2} + 8 \cdot 3 n \cdot n^{2} + 8 n (- 7 n) + 8 n \cdot 1 + 16 (3 n^{2}) + 16 (- 7 n) + 16 \cdot 1$

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

Move $n^{2}$ .

$3 n^{4} - 7 n^{3} + n^{2} + 8 \cdot 3 (n^{2} n) + 8 n (- 7 n) + 8 n \cdot 1 + 16 (3 n^{2}) + 16 (- 7 n) + 16 \cdot 1$

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

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

$3 n^{4} - 7 n^{3} + n^{2} + 8 \cdot 3 (n^{2} n^{1}) + 8 n (- 7 n) + 8 n \cdot 1 + 16 (3 n^{2}) + 16 (- 7 n) + 16 \cdot 1$

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

$3 n^{4} - 7 n^{3} + n^{2} + 8 \cdot 3 n^{2 + 1} + 8 n (- 7 n) + 8 n \cdot 1 + 16 (3 n^{2}) + 16 (- 7 n) + 16 \cdot 1$

Add $2$ and $1$ .

$3 n^{4} - 7 n^{3} + n^{2} + 8 \cdot 3 n^{3} + 8 n (- 7 n) + 8 n \cdot 1 + 16 (3 n^{2}) + 16 (- 7 n) + 16 \cdot 1$

Multiply $8$ by $3$ .

$3 n^{4} - 7 n^{3} + n^{2} + 24 n^{3} + 8 n (- 7 n) + 8 n \cdot 1 + 16 (3 n^{2}) + 16 (- 7 n) + 16 \cdot 1$

Rewrite using the commutative property of multiplication.

$3 n^{4} - 7 n^{3} + n^{2} + 24 n^{3} + 8 \cdot - 7 n \cdot n + 8 n \cdot 1 + 16 (3 n^{2}) + 16 (- 7 n) + 16 \cdot 1$

Multiply $n$ by $n$ by adding the exponents.

Move $n$ .

$3 n^{4} - 7 n^{3} + n^{2} + 24 n^{3} + 8 \cdot - 7 (n \cdot n) + 8 n \cdot 1 + 16 (3 n^{2}) + 16 (- 7 n) + 16 \cdot 1$

Multiply $n$ by $n$ .

$3 n^{4} - 7 n^{3} + n^{2} + 24 n^{3} + 8 \cdot - 7 n^{2} + 8 n \cdot 1 + 16 (3 n^{2}) + 16 (- 7 n) + 16 \cdot 1$

Multiply $8$ by $- 7$ .

$3 n^{4} - 7 n^{3} + n^{2} + 24 n^{3} - 56 n^{2} + 8 n \cdot 1 + 16 (3 n^{2}) + 16 (- 7 n) + 16 \cdot 1$

Multiply $8$ by $1$ .

$3 n^{4} - 7 n^{3} + n^{2} + 24 n^{3} - 56 n^{2} + 8 n + 16 (3 n^{2}) + 16 (- 7 n) + 16 \cdot 1$

Multiply $3$ by $16$ .

$3 n^{4} - 7 n^{3} + n^{2} + 24 n^{3} - 56 n^{2} + 8 n + 48 n^{2} + 16 (- 7 n) + 16 \cdot 1$

Multiply $- 7$ by $16$ .

$3 n^{4} - 7 n^{3} + n^{2} + 24 n^{3} - 56 n^{2} + 8 n + 48 n^{2} - 112 n + 16 \cdot 1$

Multiply $16$ by $1$ .

$3 n^{4} - 7 n^{3} + n^{2} + 24 n^{3} - 56 n^{2} + 8 n + 48 n^{2} - 112 n + 16$

Simplify by adding terms.

Add $- 7 n^{3}$ and $24 n^{3}$ .

$3 n^{4} + 17 n^{3} + n^{2} - 56 n^{2} + 8 n + 48 n^{2} - 112 n + 16$

Subtract $56 n^{2}$ from $n^{2}$ .

$3 n^{4} + 17 n^{3} - 55 n^{2} + 8 n + 48 n^{2} - 112 n + 16$

Add $- 55 n^{2}$ and $48 n^{2}$ .

$3 n^{4} + 17 n^{3} - 7 n^{2} + 8 n - 112 n + 16$

Subtract $112 n$ from $8 n$ .

$3 n^{4} + 17 n^{3} - 7 n^{2} - 104 n + 16$

Simplify (n+4)^2(3n^2-7n+1)

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