Simplify (2x^2+3xy-5y^2)^2

${(2 x^{2} + 3 x y - 5 y^{2})}^{2}$

Step 1

Rewrite ${(2 x^{2} + 3 x y - 5 y^{2})}^{2}$ as $(2 x^{2} + 3 x y - 5 y^{2}) (2 x^{2} + 3 x y - 5 y^{2})$ .

$(2 x^{2} + 3 x y - 5 y^{2}) (2 x^{2} + 3 x y - 5 y^{2})$

Step 2

Expand $(2 x^{2} + 3 x y - 5 y^{2}) (2 x^{2} + 3 x y - 5 y^{2})$ by multiplying each term in the first expression by each term in the second expression.

$2 x^{2} (2 x^{2}) + 2 x^{2} (3 x y) + 2 x^{2} (- 5 y^{2}) + 3 x y (2 x^{2}) + 3 x y (3 x y) + 3 x y (- 5 y^{2}) - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

Step 3

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x^{2} x^{2} + 2 x^{2} (3 x y) + 2 x^{2} (- 5 y^{2}) + 3 x y (2 x^{2}) + 3 x y (3 x y) + 3 x y (- 5 y^{2}) - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

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

Move $x^{2}$ .

$2 \cdot 2 (x^{2} x^{2}) + 2 x^{2} (3 x y) + 2 x^{2} (- 5 y^{2}) + 3 x y (2 x^{2}) + 3 x y (3 x y) + 3 x y (- 5 y^{2}) - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

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

$2 \cdot 2 x^{2 + 2} + 2 x^{2} (3 x y) + 2 x^{2} (- 5 y^{2}) + 3 x y (2 x^{2}) + 3 x y (3 x y) + 3 x y (- 5 y^{2}) - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

Add $2$ and $2$ .

$2 \cdot 2 x^{4} + 2 x^{2} (3 x y) + 2 x^{2} (- 5 y^{2}) + 3 x y (2 x^{2}) + 3 x y (3 x y) + 3 x y (- 5 y^{2}) - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

Multiply $2$ by $2$ .

$4 x^{4} + 2 x^{2} (3 x y) + 2 x^{2} (- 5 y^{2}) + 3 x y (2 x^{2}) + 3 x y (3 x y) + 3 x y (- 5 y^{2}) - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

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

Move $x$ .

$4 x^{4} + 2 (x \cdot x^{2}) (3 y) + 2 x^{2} (- 5 y^{2}) + 3 x y (2 x^{2}) + 3 x y (3 x y) + 3 x y (- 5 y^{2}) - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

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

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

$4 x^{4} + 2 (x^{1} x^{2}) (3 y) + 2 x^{2} (- 5 y^{2}) + 3 x y (2 x^{2}) + 3 x y (3 x y) + 3 x y (- 5 y^{2}) - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

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

$4 x^{4} + 2 x^{1 + 2} (3 y) + 2 x^{2} (- 5 y^{2}) + 3 x y (2 x^{2}) + 3 x y (3 x y) + 3 x y (- 5 y^{2}) - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

Add $1$ and $2$ .

$4 x^{4} + 2 x^{3} (3 y) + 2 x^{2} (- 5 y^{2}) + 3 x y (2 x^{2}) + 3 x y (3 x y) + 3 x y (- 5 y^{2}) - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

Rewrite using the commutative property of multiplication.

$4 x^{4} + 2 \cdot 3 x^{3} y + 2 x^{2} (- 5 y^{2}) + 3 x y (2 x^{2}) + 3 x y (3 x y) + 3 x y (- 5 y^{2}) - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

Multiply $2$ by $3$ .

$4 x^{4} + 6 x^{3} y + 2 x^{2} (- 5 y^{2}) + 3 x y (2 x^{2}) + 3 x y (3 x y) + 3 x y (- 5 y^{2}) - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

Rewrite using the commutative property of multiplication.

$4 x^{4} + 6 x^{3} y + 2 \cdot - 5 x^{2} y^{2} + 3 x y (2 x^{2}) + 3 x y (3 x y) + 3 x y (- 5 y^{2}) - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

Multiply $2$ by $- 5$ .

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 3 x y (2 x^{2}) + 3 x y (3 x y) + 3 x y (- 5 y^{2}) - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

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

Move $x^{2}$ .

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 3 (x^{2} x) y \cdot 2 + 3 x y (3 x y) + 3 x y (- 5 y^{2}) - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

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

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

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 3 (x^{2} x^{1}) y \cdot 2 + 3 x y (3 x y) + 3 x y (- 5 y^{2}) - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

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

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 3 x^{2 + 1} y \cdot 2 + 3 x y (3 x y) + 3 x y (- 5 y^{2}) - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

Add $2$ and $1$ .

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 3 x^{3} y \cdot 2 + 3 x y (3 x y) + 3 x y (- 5 y^{2}) - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

Multiply $2$ by $3$ .

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 6 x^{3} y + 3 x y (3 x y) + 3 x y (- 5 y^{2}) - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

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

Move $x$ .

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 6 x^{3} y + 3 (x \cdot x) y (3 y) + 3 x y (- 5 y^{2}) - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

Multiply $x$ by $x$ .

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 6 x^{3} y + 3 x^{2} y (3 y) + 3 x y (- 5 y^{2}) - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

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

Move $y$ .

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 6 x^{3} y + 3 x^{2} (y \cdot y) \cdot 3 + 3 x y (- 5 y^{2}) - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

Multiply $y$ by $y$ .

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 6 x^{3} y + 3 x^{2} y^{2} \cdot 3 + 3 x y (- 5 y^{2}) - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

Multiply $3$ by $3$ .

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 6 x^{3} y + 9 x^{2} y^{2} + 3 x y (- 5 y^{2}) - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

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

Move $y^{2}$ .

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 6 x^{3} y + 9 x^{2} y^{2} + 3 x (y^{2} y) \cdot - 5 - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

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

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

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 6 x^{3} y + 9 x^{2} y^{2} + 3 x (y^{2} y^{1}) \cdot - 5 - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

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

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 6 x^{3} y + 9 x^{2} y^{2} + 3 x y^{2 + 1} \cdot - 5 - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

Add $2$ and $1$ .

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 6 x^{3} y + 9 x^{2} y^{2} + 3 x y^{3} \cdot - 5 - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

Multiply $- 5$ by $3$ .

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 6 x^{3} y + 9 x^{2} y^{2} - 15 x y^{3} - 5 y^{2} (2 x^{2}) - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

Rewrite using the commutative property of multiplication.

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 6 x^{3} y + 9 x^{2} y^{2} - 15 x y^{3} - 5 \cdot 2 y^{2} x^{2} - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

Multiply $- 5$ by $2$ .

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 6 x^{3} y + 9 x^{2} y^{2} - 15 x y^{3} - 10 y^{2} x^{2} - 5 y^{2} (3 x y) - 5 y^{2} (- 5 y^{2})$

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

Move $y$ .

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 6 x^{3} y + 9 x^{2} y^{2} - 15 x y^{3} - 10 y^{2} x^{2} - 5 (y \cdot y^{2}) (3 x) - 5 y^{2} (- 5 y^{2})$

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

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

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 6 x^{3} y + 9 x^{2} y^{2} - 15 x y^{3} - 10 y^{2} x^{2} - 5 (y^{1} y^{2}) (3 x) - 5 y^{2} (- 5 y^{2})$

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

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 6 x^{3} y + 9 x^{2} y^{2} - 15 x y^{3} - 10 y^{2} x^{2} - 5 y^{1 + 2} (3 x) - 5 y^{2} (- 5 y^{2})$

Add $1$ and $2$ .

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 6 x^{3} y + 9 x^{2} y^{2} - 15 x y^{3} - 10 y^{2} x^{2} - 5 y^{3} (3 x) - 5 y^{2} (- 5 y^{2})$

Rewrite using the commutative property of multiplication.

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 6 x^{3} y + 9 x^{2} y^{2} - 15 x y^{3} - 10 y^{2} x^{2} - 5 \cdot 3 y^{3} x - 5 y^{2} (- 5 y^{2})$

Multiply $- 5$ by $3$ .

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 6 x^{3} y + 9 x^{2} y^{2} - 15 x y^{3} - 10 y^{2} x^{2} - 15 y^{3} x - 5 y^{2} (- 5 y^{2})$

Rewrite using the commutative property of multiplication.

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 6 x^{3} y + 9 x^{2} y^{2} - 15 x y^{3} - 10 y^{2} x^{2} - 15 y^{3} x - 5 \cdot - 5 y^{2} y^{2}$

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

Move $y^{2}$ .

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 6 x^{3} y + 9 x^{2} y^{2} - 15 x y^{3} - 10 y^{2} x^{2} - 15 y^{3} x - 5 \cdot - 5 (y^{2} y^{2})$

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

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 6 x^{3} y + 9 x^{2} y^{2} - 15 x y^{3} - 10 y^{2} x^{2} - 15 y^{3} x - 5 \cdot - 5 y^{2 + 2}$

Add $2$ and $2$ .

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 6 x^{3} y + 9 x^{2} y^{2} - 15 x y^{3} - 10 y^{2} x^{2} - 15 y^{3} x - 5 \cdot - 5 y^{4}$

Multiply $- 5$ by $- 5$ .

$4 x^{4} + 6 x^{3} y - 10 x^{2} y^{2} + 6 x^{3} y + 9 x^{2} y^{2} - 15 x y^{3} - 10 y^{2} x^{2} - 15 y^{3} x + 25 y^{4}$

Simplify by adding terms.

Add $6 x^{3} y$ and $6 x^{3} y$ .

$4 x^{4} + 12 x^{3} y - 10 x^{2} y^{2} + 9 x^{2} y^{2} - 15 x y^{3} - 10 y^{2} x^{2} - 15 y^{3} x + 25 y^{4}$

Add $- 10 x^{2} y^{2}$ and $9 x^{2} y^{2}$ .

$4 x^{4} + 12 x^{3} y - x^{2} y^{2} - 15 x y^{3} - 10 y^{2} x^{2} - 15 y^{3} x + 25 y^{4}$

Step 4

Subtract $10 y^{2} x^{2}$ from $- x^{2} y^{2}$ .

Move $y^{2}$ .

$4 x^{4} + 12 x^{3} y - x^{2} y^{2} - 10 x^{2} y^{2} - 15 x y^{3} - 15 y^{3} x + 25 y^{4}$

Subtract $10 x^{2} y^{2}$ from $- x^{2} y^{2}$ .

$4 x^{4} + 12 x^{3} y - 11 x^{2} y^{2} - 15 x y^{3} - 15 y^{3} x + 25 y^{4}$

Step 5

Subtract $15 y^{3} x$ from $- 15 x y^{3}$ .

Move $y^{3}$ .

$4 x^{4} + 12 x^{3} y - 11 x^{2} y^{2} - 15 x y^{3} - 15 x y^{3} + 25 y^{4}$

Subtract $15 x y^{3}$ from $- 15 x y^{3}$ .

$4 x^{4} + 12 x^{3} y - 11 x^{2} y^{2} - 30 x y^{3} + 25 y^{4}$

Simplify (2x^2+3xy-5y^2)^2

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