Solve (5y+1)(1-5y)-(1+5y)^2

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

Evaluate

$$-10y\left(5y+1\right)$$

Solution Steps

Consider \left(5y+1\right)\left(1-5y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$1-\left(5y\right)^{2}-\left(1+5y\right)^{2}$$

Expand \left(5y\right)^{2}.

$$1-5^{2}y^{2}-\left(1+5y\right)^{2}$$

Calculate 5 to the power of 2 and get 25.

$$1-25y^{2}-\left(1+5y\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+5y\right)^{2}.

$$1-25y^{2}-\left(1+10y+25y^{2}\right)$$

To find the opposite of 1+10y+25y^{2}, find the opposite of each term.

$$1-25y^{2}-1-10y-25y^{2}$$

Subtract 1 from 1 to get 0.

$$-25y^{2}-10y-25y^{2}$$

Combine -25y^{2} and -25y^{2} to get -50y^{2}.

$$-50y^{2}-10y$$

Expand

$$-50y^{2}-10y$$

Solution Steps

Consider \left(5y+1\right)\left(1-5y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$1-\left(5y\right)^{2}-\left(1+5y\right)^{2}$$

Expand \left(5y\right)^{2}.

$$1-5^{2}y^{2}-\left(1+5y\right)^{2}$$

Calculate 5 to the power of 2 and get 25.

$$1-25y^{2}-\left(1+5y\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+5y\right)^{2}.

$$1-25y^{2}-\left(1+10y+25y^{2}\right)$$

To find the opposite of 1+10y+25y^{2}, find the opposite of each term.

$$1-25y^{2}-1-10y-25y^{2}$$

Subtract 1 from 1 to get 0.

$$-25y^{2}-10y-25y^{2}$$

Combine -25y^{2} and -25y^{2} to get -50y^{2}.

$$-50y^{2}-10y$$