Solve 3(y-z)^2-(2y+z)(-z+2y)

$$3 ( y - z ) ^ { 2 } - ( 2 y + z ) ( - z + 2 y )$$

Evaluate

$$4z^{2}-6yz-y^{2}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(y-z\right)^{2}.

$$3\left(y^{2}-2yz+z^{2}\right)-\left(2y+z\right)\left(-z+2y\right)$$

Use the distributive property to multiply 3 by y^{2}-2yz+z^{2}.

$$3y^{2}-6yz+3z^{2}-\left(2y+z\right)\left(-z+2y\right)$$

Use the distributive property to multiply 2y+z by -z+2y.

$$3y^{2}-6yz+3z^{2}-\left(2y\left(-z\right)+4y^{2}+z\left(-z\right)+2zy\right)$$

To find the opposite of 2y\left(-z\right)+4y^{2}+z\left(-z\right)+2zy, find the opposite of each term.

$$3y^{2}-6yz+3z^{2}-2y\left(-z\right)-4y^{2}-z\left(-z\right)-2zy$$

Multiply -2 and -1 to get 2.

$$3y^{2}-6yz+3z^{2}+2yz-4y^{2}-z\left(-z\right)-2zy$$

Combine -6yz and 2yz to get -4yz.

$$3y^{2}-4yz+3z^{2}-4y^{2}-z\left(-z\right)-2zy$$

Combine 3y^{2} and -4y^{2} to get -y^{2}.

$$-y^{2}-4yz+3z^{2}-z\left(-z\right)-2zy$$

Multiply -1 and -1 to get 1.

$$-y^{2}-4yz+3z^{2}+zz-2zy$$

Multiply z and z to get z^{2}.

$$-y^{2}-4yz+3z^{2}+z^{2}-2zy$$

Combine 3z^{2} and z^{2} to get 4z^{2}.

$$-y^{2}-4yz+4z^{2}-2zy$$

Combine -4yz and -2zy to get -6yz.

$$-y^{2}-6yz+4z^{2}$$

Expand

$$4z^{2}-6yz-y^{2}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(y-z\right)^{2}.

$$3\left(y^{2}-2yz+z^{2}\right)-\left(2y+z\right)\left(-z+2y\right)$$

Use the distributive property to multiply 3 by y^{2}-2yz+z^{2}.

$$3y^{2}-6yz+3z^{2}-\left(2y+z\right)\left(-z+2y\right)$$

Use the distributive property to multiply 2y+z by -z+2y.

$$3y^{2}-6yz+3z^{2}-\left(2y\left(-z\right)+4y^{2}+z\left(-z\right)+2zy\right)$$

To find the opposite of 2y\left(-z\right)+4y^{2}+z\left(-z\right)+2zy, find the opposite of each term.

$$3y^{2}-6yz+3z^{2}-2y\left(-z\right)-4y^{2}-z\left(-z\right)-2zy$$

Multiply -2 and -1 to get 2.

$$3y^{2}-6yz+3z^{2}+2yz-4y^{2}-z\left(-z\right)-2zy$$

Combine -6yz and 2yz to get -4yz.

$$3y^{2}-4yz+3z^{2}-4y^{2}-z\left(-z\right)-2zy$$

Combine 3y^{2} and -4y^{2} to get -y^{2}.

$$-y^{2}-4yz+3z^{2}-z\left(-z\right)-2zy$$

Multiply -1 and -1 to get 1.

$$-y^{2}-4yz+3z^{2}+zz-2zy$$

Multiply z and z to get z^{2}.

$$-y^{2}-4yz+3z^{2}+z^{2}-2zy$$

Combine 3z^{2} and z^{2} to get 4z^{2}.

$$-y^{2}-4yz+4z^{2}-2zy$$

Combine -4yz and -2zy to get -6yz.

$$-y^{2}-6yz+4z^{2}$$