Solve (4)^4(2a+b)(2a+b)-(a+b)^2

$$( 4 ) ^ { 4 } ( 2 a + b ) ( 2 a + b ) - ( a + b ) ^ { 2 }$$

Evaluate

$$\left(31a+15b\right)\left(33a+17b\right)$$

Solution Steps

Multiply 2a+b and 2a+b to get \left(2a+b\right)^{2}.

$$4^{4}\left(2a+b\right)^{2}-\left(a+b\right)^{2}$$

Calculate 4 to the power of 4 and get 256.

$$256\left(2a+b\right)^{2}-\left(a+b\right)^{2}$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(2a+b\right)^{2}.

$$256\left(4a^{2}+4ab+b^{2}\right)-\left(a+b\right)^{2}$$

Use the distributive property to multiply 256 by 4a^{2}+4ab+b^{2}.

$$1024a^{2}+1024ab+256b^{2}-\left(a+b\right)^{2}$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a+b\right)^{2}.

$$1024a^{2}+1024ab+256b^{2}-\left(a^{2}+2ab+b^{2}\right)$$

To find the opposite of a^{2}+2ab+b^{2}, find the opposite of each term.

$$1024a^{2}+1024ab+256b^{2}-a^{2}-2ab-b^{2}$$

Combine 1024a^{2} and -a^{2} to get 1023a^{2}.

$$1023a^{2}+1024ab+256b^{2}-2ab-b^{2}$$

Combine 1024ab and -2ab to get 1022ab.

$$1023a^{2}+1022ab+256b^{2}-b^{2}$$

Combine 256b^{2} and -b^{2} to get 255b^{2}.

$$1023a^{2}+1022ab+255b^{2}$$

Expand

$$1023a^{2}+1022ab+255b^{2}$$

Solution Steps

Multiply 2a+b and 2a+b to get \left(2a+b\right)^{2}.

$$4^{4}\left(2a+b\right)^{2}-\left(a+b\right)^{2}$$

Calculate 4 to the power of 4 and get 256.

$$256\left(2a+b\right)^{2}-\left(a+b\right)^{2}$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(2a+b\right)^{2}.

$$256\left(4a^{2}+4ab+b^{2}\right)-\left(a+b\right)^{2}$$

Use the distributive property to multiply 256 by 4a^{2}+4ab+b^{2}.

$$1024a^{2}+1024ab+256b^{2}-\left(a+b\right)^{2}$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a+b\right)^{2}.

$$1024a^{2}+1024ab+256b^{2}-\left(a^{2}+2ab+b^{2}\right)$$

To find the opposite of a^{2}+2ab+b^{2}, find the opposite of each term.

$$1024a^{2}+1024ab+256b^{2}-a^{2}-2ab-b^{2}$$

Combine 1024a^{2} and -a^{2} to get 1023a^{2}.

$$1023a^{2}+1024ab+256b^{2}-2ab-b^{2}$$

Combine 1024ab and -2ab to get 1022ab.

$$1023a^{2}+1022ab+256b^{2}-b^{2}$$

Combine 256b^{2} and -b^{2} to get 255b^{2}.

$$1023a^{2}+1022ab+255b^{2}$$