Find the Equation Given the Roots 3 , 4+2i , 1+ square root of 7

$3$ , $4 + 2 i$ , $1 + \sqrt{7}$

Roots are the points where the graph intercepts with the x-axis $(y = 0)$ .

$y = 0$ at the roots

The root at $x = 3$ was found by solving for $x$ when $x - (3) = y$ and $y = 0$ .

The factor is $x - 3$

The root at $x = 4 + 2 i$ was found by solving for $x$ when $x - (4 + 2 i) = y$ and $y = 0$ .

The factor is $x - 4 - 2 i$

The root at $x = 4 - 2 i$ was found by solving for $x$ when $x - (4 - 2 i) = y$ and $y = 0$ .

The factor is $x - 4 + 2 i$

The root at $x = 1 + \sqrt{7}$ was found by solving for $x$ when $x - (1 + \sqrt{7}) = y$ and $y = 0$ .

The factor is $x - 1 - \sqrt{7}$

Combine all the factors into a single equation.

$y = (x - 3) (x - 4 - 2 i) (x - 4 + 2 i) (x - 1 - \sqrt{7})$

Multiply all the factors to simplify the equation $y = (x - 3) (x - 4 - 2 i) (x - 4 + 2 i) (x - 1 - \sqrt{7})$ .

Expand $(x - 3) (x - 4 - 2 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x \cdot x + x \cdot - 4 + x (- 2 i) - 3 x - 3 \cdot - 4 - 3 (- 2 i)) (x - 4 + 2 i) (x - 1 - \sqrt{7})$

Simplify terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = (x^{2} + x \cdot - 4 + x (- 2 i) - 3 x - 3 \cdot - 4 - 3 (- 2 i)) (x - 4 + 2 i) (x - 1 - \sqrt{7})$

Move $- 4$ to the left of $x$ .

$y = (x^{2} - 4 \cdot x + x (- 2 i) - 3 x - 3 \cdot - 4 - 3 (- 2 i)) (x - 4 + 2 i) (x - 1 - \sqrt{7})$

Multiply $- 3$ by $- 4$ .

$y = (x^{2} - 4 x + x (- 2 i) - 3 x + 12 - 3 (- 2 i)) (x - 4 + 2 i) (x - 1 - \sqrt{7})$

Multiply $- 2$ by $- 3$ .

$y = (x^{2} - 4 x + x (- 2 i) - 3 x + 12 + 6 i) (x - 4 + 2 i) (x - 1 - \sqrt{7})$

Subtract $3 x$ from $- 4 x$ .

$y = (x^{2} + x (- 2 i) - 7 x + 12 + 6 i) (x - 4 + 2 i) (x - 1 - \sqrt{7})$

Expand $(x^{2} + x (- 2 i) - 7 x + 12 + 6 i) (x - 4 + 2 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x^{2} x + x^{2} \cdot - 4 + x^{2} (2 i) + x (- 2 i) x + x (- 2 i) \cdot - 4 + x (- 2 i) (2 i) - 7 x \cdot x - 7 x \cdot - 4 - 7 x (2 i) + 12 x + 12 \cdot - 4 + 12 (2 i) + 6 i x + 6 i \cdot - 4 + 6 i (2 i)) (x - 1 - \sqrt{7})$

Simplify terms.

Simplify each term.

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

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

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

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

$y = (x^{2 + 1} + x^{2} \cdot - 4 + x^{2} (2 i) + x (- 2 i) x + x (- 2 i) \cdot - 4 + x (- 2 i) (2 i) - 7 x \cdot x - 7 x \cdot - 4 - 7 x (2 i) + 12 x + 12 \cdot - 4 + 12 (2 i) + 6 i x + 6 i \cdot - 4 + 6 i (2 i)) (x - 1 - \sqrt{7})$

Add $2$ and $1$ .

$y = (x^{3} + x^{2} \cdot - 4 + x^{2} (2 i) + x (- 2 i) x + x (- 2 i) \cdot - 4 + x (- 2 i) (2 i) - 7 x \cdot x - 7 x \cdot - 4 - 7 x (2 i) + 12 x + 12 \cdot - 4 + 12 (2 i) + 6 i x + 6 i \cdot - 4 + 6 i (2 i)) (x - 1 - \sqrt{7})$

Move $- 4$ to the left of $x^{2}$ .

$y = (x^{3} - 4 \cdot x^{2} + x^{2} (2 i) + x (- 2 i) x + x (- 2 i) \cdot - 4 + x (- 2 i) (2 i) - 7 x \cdot x - 7 x \cdot - 4 - 7 x (2 i) + 12 x + 12 \cdot - 4 + 12 (2 i) + 6 i x + 6 i \cdot - 4 + 6 i (2 i)) (x - 1 - \sqrt{7})$

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

$y = (x^{3} - 4 x^{2} + 2 \cdot (x^{2} i) + x (- 2 i) x + x (- 2 i) \cdot - 4 + x (- 2 i) (2 i) - 7 x \cdot x - 7 x \cdot - 4 - 7 x (2 i) + 12 x + 12 \cdot - 4 + 12 (2 i) + 6 i x + 6 i \cdot - 4 + 6 i (2 i)) (x - 1 - \sqrt{7})$

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

Move $x$ .

$y = (x^{3} - 4 x^{2} + 2 x^{2} i + x \cdot x (- 2 i) + x (- 2 i) \cdot - 4 + x (- 2 i) (2 i) - 7 x \cdot x - 7 x \cdot - 4 - 7 x (2 i) + 12 x + 12 \cdot - 4 + 12 (2 i) + 6 i x + 6 i \cdot - 4 + 6 i (2 i)) (x - 1 - \sqrt{7})$

Multiply $x$ by $x$ .

$y = (x^{3} - 4 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + x (- 2 i) \cdot - 4 + x (- 2 i) (2 i) - 7 x \cdot x - 7 x \cdot - 4 - 7 x (2 i) + 12 x + 12 \cdot - 4 + 12 (2 i) + 6 i x + 6 i \cdot - 4 + 6 i (2 i)) (x - 1 - \sqrt{7})$

Multiply $- 4$ by $- 2$ .

$y = (x^{3} - 4 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + x (8 i) + x (- 2 i) (2 i) - 7 x \cdot x - 7 x \cdot - 4 - 7 x (2 i) + 12 x + 12 \cdot - 4 + 12 (2 i) + 6 i x + 6 i \cdot - 4 + 6 i (2 i)) (x - 1 - \sqrt{7})$

Move $8$ to the left of $x$ .

$y = (x^{3} - 4 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 8 \cdot (x i) + x (- 2 i) (2 i) - 7 x \cdot x - 7 x \cdot - 4 - 7 x (2 i) + 12 x + 12 \cdot - 4 + 12 (2 i) + 6 i x + 6 i \cdot - 4 + 6 i (2 i)) (x - 1 - \sqrt{7})$

Multiply $x (- 2 i) (2 i)$ .

Multiply $2$ by $- 2$ .

$y = (x^{3} - 4 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 8 x i + x (- 4 i) i - 7 x \cdot x - 7 x \cdot - 4 - 7 x (2 i) + 12 x + 12 \cdot - 4 + 12 (2 i) + 6 i x + 6 i \cdot - 4 + 6 i (2 i)) (x - 1 - \sqrt{7})$

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

$y = (x^{3} - 4 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 8 x i + x (- 4 (i i)) - 7 x \cdot x - 7 x \cdot - 4 - 7 x (2 i) + 12 x + 12 \cdot - 4 + 12 (2 i) + 6 i x + 6 i \cdot - 4 + 6 i (2 i)) (x - 1 - \sqrt{7})$

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

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

$y = (x^{3} - 4 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 8 x i + x (- 4 i^{1 + 1}) - 7 x \cdot x - 7 x \cdot - 4 - 7 x (2 i) + 12 x + 12 \cdot - 4 + 12 (2 i) + 6 i x + 6 i \cdot - 4 + 6 i (2 i)) (x - 1 - \sqrt{7})$

Add $1$ and $1$ .

$y = (x^{3} - 4 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 8 x i + x (- 4 i^{2}) - 7 x \cdot x - 7 x \cdot - 4 - 7 x (2 i) + 12 x + 12 \cdot - 4 + 12 (2 i) + 6 i x + 6 i \cdot - 4 + 6 i (2 i)) (x - 1 - \sqrt{7})$

Rewrite $i^{2}$ as $- 1$ .

$y = (x^{3} - 4 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 8 x i + x (- 4 \cdot - 1) - 7 x \cdot x - 7 x \cdot - 4 - 7 x (2 i) + 12 x + 12 \cdot - 4 + 12 (2 i) + 6 i x + 6 i \cdot - 4 + 6 i (2 i)) (x - 1 - \sqrt{7})$

Multiply $- 4$ by $- 1$ .

$y = (x^{3} - 4 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 8 x i + x \cdot 4 - 7 x \cdot x - 7 x \cdot - 4 - 7 x (2 i) + 12 x + 12 \cdot - 4 + 12 (2 i) + 6 i x + 6 i \cdot - 4 + 6 i (2 i)) (x - 1 - \sqrt{7})$

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

$y = (x^{3} - 4 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 8 x i + 4 \cdot x - 7 x \cdot x - 7 x \cdot - 4 - 7 x (2 i) + 12 x + 12 \cdot - 4 + 12 (2 i) + 6 i x + 6 i \cdot - 4 + 6 i (2 i)) (x - 1 - \sqrt{7})$

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

Move $x$ .

$y = (x^{3} - 4 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 8 x i + 4 x - 7 (x \cdot x) - 7 x \cdot - 4 - 7 x (2 i) + 12 x + 12 \cdot - 4 + 12 (2 i) + 6 i x + 6 i \cdot - 4 + 6 i (2 i)) (x - 1 - \sqrt{7})$

Multiply $x$ by $x$ .

$y = (x^{3} - 4 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 8 x i + 4 x - 7 x^{2} - 7 x \cdot - 4 - 7 x (2 i) + 12 x + 12 \cdot - 4 + 12 (2 i) + 6 i x + 6 i \cdot - 4 + 6 i (2 i)) (x - 1 - \sqrt{7})$

Multiply $- 4$ by $- 7$ .

$y = (x^{3} - 4 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 8 x i + 4 x - 7 x^{2} + 28 x - 7 x (2 i) + 12 x + 12 \cdot - 4 + 12 (2 i) + 6 i x + 6 i \cdot - 4 + 6 i (2 i)) (x - 1 - \sqrt{7})$

Multiply $2$ by $- 7$ .

$y = (x^{3} - 4 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 8 x i + 4 x - 7 x^{2} + 28 x - 14 x i + 12 x + 12 \cdot - 4 + 12 (2 i) + 6 i x + 6 i \cdot - 4 + 6 i (2 i)) (x - 1 - \sqrt{7})$

Multiply $12$ by $- 4$ .

$y = (x^{3} - 4 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 8 x i + 4 x - 7 x^{2} + 28 x - 14 x i + 12 x - 48 + 12 (2 i) + 6 i x + 6 i \cdot - 4 + 6 i (2 i)) (x - 1 - \sqrt{7})$

Multiply $2$ by $12$ .

$y = (x^{3} - 4 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 8 x i + 4 x - 7 x^{2} + 28 x - 14 x i + 12 x - 48 + 24 i + 6 i x + 6 i \cdot - 4 + 6 i (2 i)) (x - 1 - \sqrt{7})$

Multiply $- 4$ by $6$ .

$y = (x^{3} - 4 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 8 x i + 4 x - 7 x^{2} + 28 x - 14 x i + 12 x - 48 + 24 i + 6 i x - 24 i + 6 i (2 i)) (x - 1 - \sqrt{7})$

Multiply $6 i (2 i)$ .

Multiply $2$ by $6$ .

$y = (x^{3} - 4 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 8 x i + 4 x - 7 x^{2} + 28 x - 14 x i + 12 x - 48 + 24 i + 6 i x - 24 i + 12 i i) (x - 1 - \sqrt{7})$

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

$y = (x^{3} - 4 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 8 x i + 4 x - 7 x^{2} + 28 x - 14 x i + 12 x - 48 + 24 i + 6 i x - 24 i + 12 (i i)) (x - 1 - \sqrt{7})$

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

$y = (x^{3} - 4 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 8 x i + 4 x - 7 x^{2} + 28 x - 14 x i + 12 x - 48 + 24 i + 6 i x - 24 i + 12 (i i)) (x - 1 - \sqrt{7})$

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

$y = (x^{3} - 4 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 8 x i + 4 x - 7 x^{2} + 28 x - 14 x i + 12 x - 48 + 24 i + 6 i x - 24 i + 12 i^{1 + 1}) (x - 1 - \sqrt{7})$

Add $1$ and $1$ .

$y = (x^{3} - 4 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 8 x i + 4 x - 7 x^{2} + 28 x - 14 x i + 12 x - 48 + 24 i + 6 i x - 24 i + 12 i^{2}) (x - 1 - \sqrt{7})$

Rewrite $i^{2}$ as $- 1$ .

$y = (x^{3} - 4 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 8 x i + 4 x - 7 x^{2} + 28 x - 14 x i + 12 x - 48 + 24 i + 6 i x - 24 i + 12 \cdot - 1) (x - 1 - \sqrt{7})$

Multiply $12$ by $- 1$ .

$y = (x^{3} - 4 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 8 x i + 4 x - 7 x^{2} + 28 x - 14 x i + 12 x - 48 + 24 i + 6 i x - 24 i - 12) (x - 1 - \sqrt{7})$

Simplify by adding terms.

Combine the opposite terms in $x^{3} - 4 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 8 x i + 4 x - 7 x^{2} + 28 x - 14 x i + 12 x - 48 + 24 i + 6 i x - 24 i - 12$ .

Reorder the factors in the terms $2 x^{2} i$ and $x^{2} (- 2 i)$ .

$y = (x^{3} - 4 x^{2} + 2 i x^{2} - 2 i x^{2} + 8 x i + 4 x - 7 x^{2} + 28 x - 14 x i + 12 x - 48 + 24 i + 6 i x - 24 i - 12) (x - 1 - \sqrt{7})$

Subtract $2 i x^{2}$ from $2 i x^{2}$ .

$y = (x^{3} - 4 x^{2} + 0 + 8 x i + 4 x - 7 x^{2} + 28 x - 14 x i + 12 x - 48 + 24 i + 6 i x - 24 i - 12) (x - 1 - \sqrt{7})$

Add $x^{3} - 4 x^{2}$ and $0$ .

$y = (x^{3} - 4 x^{2} + 8 x i + 4 x - 7 x^{2} + 28 x - 14 x i + 12 x - 48 + 24 i + 6 i x - 24 i - 12) (x - 1 - \sqrt{7})$

Subtract $24 i$ from $24 i$ .

$y = (x^{3} - 4 x^{2} + 8 x i + 4 x - 7 x^{2} + 28 x - 14 x i + 12 x - 48 + 6 i x + 0 - 12) (x - 1 - \sqrt{7})$

Add $x^{3} - 4 x^{2} + 8 x i + 4 x - 7 x^{2} + 28 x - 14 x i + 12 x - 48 + 6 i x$ and $0$ .

$y = (x^{3} - 4 x^{2} + 8 x i + 4 x - 7 x^{2} + 28 x - 14 x i + 12 x - 48 + 6 i x - 12) (x - 1 - \sqrt{7})$

Subtract $7 x^{2}$ from $- 4 x^{2}$ .

$y = (x^{3} - 11 x^{2} + 8 x i + 4 x + 28 x - 14 x i + 12 x - 48 + 6 i x - 12) (x - 1 - \sqrt{7})$

Subtract $14 x i$ from $8 x i$ .

$y = (x^{3} - 11 x^{2} - 6 x i + 4 x + 28 x + 12 x - 48 + 6 i x - 12) (x - 1 - \sqrt{7})$

Combine the opposite terms in $x^{3} - 11 x^{2} - 6 x i + 4 x + 28 x + 12 x - 48 + 6 i x - 12$ .

Reorder the factors in the terms $- 6 x i$ and $6 i x$ .

$y = (x^{3} - 11 x^{2} - 6 i x + 4 x + 28 x + 12 x - 48 + 6 i x - 12) (x - 1 - \sqrt{7})$

Add $- 6 i x$ and $6 i x$ .

$y = (x^{3} - 11 x^{2} + 4 x + 28 x + 12 x - 48 + 0 - 12) (x - 1 - \sqrt{7})$

Add $x^{3} - 11 x^{2} + 4 x + 28 x + 12 x - 48$ and $0$ .

$y = (x^{3} - 11 x^{2} + 4 x + 28 x + 12 x - 48 - 12) (x - 1 - \sqrt{7})$

Add $4 x$ and $28 x$ .

$y = (x^{3} - 11 x^{2} + 32 x + 12 x - 48 - 12) (x - 1 - \sqrt{7})$

Add $32 x$ and $12 x$ .

$y = (x^{3} - 11 x^{2} + 44 x - 48 - 12) (x - 1 - \sqrt{7})$

Subtract $12$ from $- 48$ .

$y = (x^{3} - 11 x^{2} + 44 x - 60) (x - 1 - \sqrt{7})$

Expand $(x^{3} - 11 x^{2} + 44 x - 60) (x - 1 - \sqrt{7})$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{3} x + x^{3} \cdot - 1 + x^{3} (- \sqrt{7}) - 11 x^{2} x - 11 x^{2} \cdot - 1 - 11 x^{2} (- \sqrt{7}) + 44 x \cdot x + 44 x \cdot - 1 + 44 x (- \sqrt{7}) - 60 x - 60 \cdot - 1 - 60 (- \sqrt{7})$

Simplify terms.

Simplify each term.

Multiply $x^{3}$ by $x$ by adding the exponents.

Multiply $x^{3}$ by $x$ .

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

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

$y = x^{3 + 1} + x^{3} \cdot - 1 + x^{3} (- \sqrt{7}) - 11 x^{2} x - 11 x^{2} \cdot - 1 - 11 x^{2} (- \sqrt{7}) + 44 x \cdot x + 44 x \cdot - 1 + 44 x (- \sqrt{7}) - 60 x - 60 \cdot - 1 - 60 (- \sqrt{7})$

Add $3$ and $1$ .

$y = x^{4} + x^{3} \cdot - 1 + x^{3} (- \sqrt{7}) - 11 x^{2} x - 11 x^{2} \cdot - 1 - 11 x^{2} (- \sqrt{7}) + 44 x \cdot x + 44 x \cdot - 1 + 44 x (- \sqrt{7}) - 60 x - 60 \cdot - 1 - 60 (- \sqrt{7})$

Move $- 1$ to the left of $x^{3}$ .

$y = x^{4} - 1 \cdot x^{3} + x^{3} (- \sqrt{7}) - 11 x^{2} x - 11 x^{2} \cdot - 1 - 11 x^{2} (- \sqrt{7}) + 44 x \cdot x + 44 x \cdot - 1 + 44 x (- \sqrt{7}) - 60 x - 60 \cdot - 1 - 60 (- \sqrt{7})$

Rewrite $- 1 x^{3}$ as $- x^{3}$ .

$y = x^{4} - x^{3} + x^{3} (- \sqrt{7}) - 11 x^{2} x - 11 x^{2} \cdot - 1 - 11 x^{2} (- \sqrt{7}) + 44 x \cdot x + 44 x \cdot - 1 + 44 x (- \sqrt{7}) - 60 x - 60 \cdot - 1 - 60 (- \sqrt{7})$

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

Move $x$ .

$y = x^{4} - x^{3} + x^{3} (- \sqrt{7}) - 11 (x \cdot x^{2}) - 11 x^{2} \cdot - 1 - 11 x^{2} (- \sqrt{7}) + 44 x \cdot x + 44 x \cdot - 1 + 44 x (- \sqrt{7}) - 60 x - 60 \cdot - 1 - 60 (- \sqrt{7})$

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

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

$y = x^{4} - x^{3} + x^{3} (- \sqrt{7}) - 11 (x \cdot x^{2}) - 11 x^{2} \cdot - 1 - 11 x^{2} (- \sqrt{7}) + 44 x \cdot x + 44 x \cdot - 1 + 44 x (- \sqrt{7}) - 60 x - 60 \cdot - 1 - 60 (- \sqrt{7})$

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

$y = x^{4} - x^{3} + x^{3} (- \sqrt{7}) - 11 x^{1 + 2} - 11 x^{2} \cdot - 1 - 11 x^{2} (- \sqrt{7}) + 44 x \cdot x + 44 x \cdot - 1 + 44 x (- \sqrt{7}) - 60 x - 60 \cdot - 1 - 60 (- \sqrt{7})$

Add $1$ and $2$ .

$y = x^{4} - x^{3} + x^{3} (- \sqrt{7}) - 11 x^{3} - 11 x^{2} \cdot - 1 - 11 x^{2} (- \sqrt{7}) + 44 x \cdot x + 44 x \cdot - 1 + 44 x (- \sqrt{7}) - 60 x - 60 \cdot - 1 - 60 (- \sqrt{7})$

Multiply $- 1$ by $- 11$ .

$y = x^{4} - x^{3} + x^{3} (- \sqrt{7}) - 11 x^{3} + 11 x^{2} - 11 x^{2} (- \sqrt{7}) + 44 x \cdot x + 44 x \cdot - 1 + 44 x (- \sqrt{7}) - 60 x - 60 \cdot - 1 - 60 (- \sqrt{7})$

Multiply $- 1$ by $- 11$ .

$y = x^{4} - x^{3} + x^{3} (- \sqrt{7}) - 11 x^{3} + 11 x^{2} + 11 x^{2} \sqrt{7} + 44 x \cdot x + 44 x \cdot - 1 + 44 x (- \sqrt{7}) - 60 x - 60 \cdot - 1 - 60 (- \sqrt{7})$

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

Move $x$ .

$y = x^{4} - x^{3} + x^{3} (- \sqrt{7}) - 11 x^{3} + 11 x^{2} + 11 x^{2} \sqrt{7} + 44 (x \cdot x) + 44 x \cdot - 1 + 44 x (- \sqrt{7}) - 60 x - 60 \cdot - 1 - 60 (- \sqrt{7})$

Multiply $x$ by $x$ .

$y = x^{4} - x^{3} + x^{3} (- \sqrt{7}) - 11 x^{3} + 11 x^{2} + 11 x^{2} \sqrt{7} + 44 x^{2} + 44 x \cdot - 1 + 44 x (- \sqrt{7}) - 60 x - 60 \cdot - 1 - 60 (- \sqrt{7})$

Multiply $- 1$ by $44$ .

$y = x^{4} - x^{3} + x^{3} (- \sqrt{7}) - 11 x^{3} + 11 x^{2} + 11 x^{2} \sqrt{7} + 44 x^{2} - 44 x + 44 x (- \sqrt{7}) - 60 x - 60 \cdot - 1 - 60 (- \sqrt{7})$

Multiply $- 1$ by $44$ .

$y = x^{4} - x^{3} + x^{3} (- \sqrt{7}) - 11 x^{3} + 11 x^{2} + 11 x^{2} \sqrt{7} + 44 x^{2} - 44 x - 44 x \sqrt{7} - 60 x - 60 \cdot - 1 - 60 (- \sqrt{7})$

Multiply $- 60$ by $- 1$ .

$y = x^{4} - x^{3} + x^{3} (- \sqrt{7}) - 11 x^{3} + 11 x^{2} + 11 x^{2} \sqrt{7} + 44 x^{2} - 44 x - 44 x \sqrt{7} - 60 x + 60 - 60 (- \sqrt{7})$

Multiply $- 1$ by $- 60$ .

$y = x^{4} - x^{3} + x^{3} (- \sqrt{7}) - 11 x^{3} + 11 x^{2} + 11 x^{2} \sqrt{7} + 44 x^{2} - 44 x - 44 x \sqrt{7} - 60 x + 60 + 60 \sqrt{7}$

Simplify by adding terms.

Subtract $11 x^{3}$ from $- x^{3}$ .

$y = x^{4} - 12 x^{3} + x^{3} (- \sqrt{7}) + 11 x^{2} + 11 x^{2} \sqrt{7} + 44 x^{2} - 44 x - 44 x \sqrt{7} - 60 x + 60 + 60 \sqrt{7}$

Add $11 x^{2}$ and $44 x^{2}$ .

$y = x^{4} - 12 x^{3} + x^{3} (- \sqrt{7}) + 55 x^{2} + 11 x^{2} \sqrt{7} - 44 x - 44 x \sqrt{7} - 60 x + 60 + 60 \sqrt{7}$

Subtract $60 x$ from $- 44 x$ .

$y = x^{4} - 12 x^{3} + x^{3} (- \sqrt{7}) + 55 x^{2} + 11 x^{2} \sqrt{7} - 44 x \sqrt{7} - 104 x + 60 + 60 \sqrt{7}$

Reorder factors in $x^{4} - 12 x^{3} + x^{3} (- \sqrt{7}) + 55 x^{2} + 11 x^{2} \sqrt{7} - 44 x \sqrt{7} - 104 x + 60 + 60 \sqrt{7}$ .

$y = x^{4} - 12 x^{3} - x^{3} \sqrt{7} + 55 x^{2} + 11 x^{2} \sqrt{7} - 44 x \sqrt{7} - 104 x + 60 + 60 \sqrt{7}$

Find the Equation Given the Roots 3 , 4+2i , 1+ square root of 7

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