Solve 4(a-b)^2-2(a-b)+5(a-b)+3(a-b)^2

$$4 ( a - b ) ^ { 2 } - 2 ( a - b ) + 5 ( a - b ) + 3 ( a - b ) ^ { 2 }$$

Evaluate

$$\left(a-b\right)\left(7a-7b+3\right)$$

Solution Steps

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

$$4\left(a^{2}-2ab+b^{2}\right)-2\left(a-b\right)+5\left(a-b\right)+3\left(a-b\right)^{2}$$

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

$$4a^{2}-8ab+4b^{2}-2\left(a-b\right)+5\left(a-b\right)+3\left(a-b\right)^{2}$$

Use the distributive property to multiply -2 by a-b.

$$4a^{2}-8ab+4b^{2}-2a+2b+5\left(a-b\right)+3\left(a-b\right)^{2}$$

Use the distributive property to multiply 5 by a-b.

$$4a^{2}-8ab+4b^{2}-2a+2b+5a-5b+3\left(a-b\right)^{2}$$

Combine -2a and 5a to get 3a.

$$4a^{2}-8ab+4b^{2}+3a+2b-5b+3\left(a-b\right)^{2}$$

Combine 2b and -5b to get -3b.

$$4a^{2}-8ab+4b^{2}+3a-3b+3\left(a-b\right)^{2}$$

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

$$4a^{2}-8ab+4b^{2}+3a-3b+3\left(a^{2}-2ab+b^{2}\right)$$

Use the distributive property to multiply 3 by a^{2}-2ab+b^{2}.

$$4a^{2}-8ab+4b^{2}+3a-3b+3a^{2}-6ab+3b^{2}$$

Combine 4a^{2} and 3a^{2} to get 7a^{2}.

$$7a^{2}-8ab+4b^{2}+3a-3b-6ab+3b^{2}$$

Combine -8ab and -6ab to get -14ab.

$$7a^{2}-14ab+4b^{2}+3a-3b+3b^{2}$$

Combine 4b^{2} and 3b^{2} to get 7b^{2}.

$$7a^{2}-14ab+7b^{2}+3a-3b$$

Expand

$$7a^{2}-14ab+3a+7b^{2}-3b$$

Solution Steps

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

$$4\left(a^{2}-2ab+b^{2}\right)-2\left(a-b\right)+5\left(a-b\right)+3\left(a-b\right)^{2}$$

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

$$4a^{2}-8ab+4b^{2}-2\left(a-b\right)+5\left(a-b\right)+3\left(a-b\right)^{2}$$

Use the distributive property to multiply -2 by a-b.

$$4a^{2}-8ab+4b^{2}-2a+2b+5\left(a-b\right)+3\left(a-b\right)^{2}$$

Use the distributive property to multiply 5 by a-b.

$$4a^{2}-8ab+4b^{2}-2a+2b+5a-5b+3\left(a-b\right)^{2}$$

Combine -2a and 5a to get 3a.

$$4a^{2}-8ab+4b^{2}+3a+2b-5b+3\left(a-b\right)^{2}$$

Combine 2b and -5b to get -3b.

$$4a^{2}-8ab+4b^{2}+3a-3b+3\left(a-b\right)^{2}$$

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

$$4a^{2}-8ab+4b^{2}+3a-3b+3\left(a^{2}-2ab+b^{2}\right)$$

Use the distributive property to multiply 3 by a^{2}-2ab+b^{2}.

$$4a^{2}-8ab+4b^{2}+3a-3b+3a^{2}-6ab+3b^{2}$$

Combine 4a^{2} and 3a^{2} to get 7a^{2}.

$$7a^{2}-8ab+4b^{2}+3a-3b-6ab+3b^{2}$$

Combine -8ab and -6ab to get -14ab.

$$7a^{2}-14ab+4b^{2}+3a-3b+3b^{2}$$

Combine 4b^{2} and 3b^{2} to get 7b^{2}.

$$7a^{2}-14ab+7b^{2}+3a-3b$$