Solve left(3left(1+bright)^2-4left(1+bright)+1right)^2

$${ \left(3 { \left(1+b \right) }^{ 2 } -4 \left( 1+b \right) +1 \right) }^{ 2 }$$

Evaluate

$$\left(b\left(3b+2\right)\right)^{2}$$

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(1+b\right)^{2}.

$$\left(3\left(1+2b+b^{2}\right)-4\left(1+b\right)+1\right)^{2}$$

Use the distributive property to multiply 3 by 1+2b+b^{2}.

$$\left(3+6b+3b^{2}-4\left(1+b\right)+1\right)^{2}$$

Use the distributive property to multiply -4 by 1+b.

$$\left(3+6b+3b^{2}-4-4b+1\right)^{2}$$

Subtract 4 from 3 to get -1.

$$\left(-1+6b+3b^{2}-4b+1\right)^{2}$$

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

$$\left(-1+2b+3b^{2}+1\right)^{2}$$

Add -1 and 1 to get 0.

$$\left(2b+3b^{2}\right)^{2}$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(2b+3b^{2}\right)^{2}.

$$4b^{2}+12bb^{2}+9\left(b^{2}\right)^{2}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$4b^{2}+12b^{3}+9\left(b^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$4b^{2}+12b^{3}+9b^{4}$$

Expand

$$9b^{4}+12b^{3}+4b^{2}$$

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(1+b\right)^{2}.

$$\left(3\left(1+2b+b^{2}\right)-4\left(1+b\right)+1\right)^{2}$$

Use the distributive property to multiply 3 by 1+2b+b^{2}.

$$\left(3+6b+3b^{2}-4\left(1+b\right)+1\right)^{2}$$

Use the distributive property to multiply -4 by 1+b.

$$\left(3+6b+3b^{2}-4-4b+1\right)^{2}$$

Subtract 4 from 3 to get -1.

$$\left(-1+6b+3b^{2}-4b+1\right)^{2}$$

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

$$\left(-1+2b+3b^{2}+1\right)^{2}$$

Add -1 and 1 to get 0.

$$\left(2b+3b^{2}\right)^{2}$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(2b+3b^{2}\right)^{2}.

$$4b^{2}+12bb^{2}+9\left(b^{2}\right)^{2}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$4b^{2}+12b^{3}+9\left(b^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$4b^{2}+12b^{3}+9b^{4}$$