Solve (m^{3}-2m^{2})^2+2(m^4-2m^3)+m^2

$$( m ^ { 3 } - 2 m ^ { 2 } ) ^ { 2 } + 2 ( m ^ { 4 } - 2 m ^ { 3 } ) + m ^ { 2 }$$

Evaluate

$$m^{2}\left(m-1\right)^{4}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(m^{3}-2m^{2}\right)^{2}.

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

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

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

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

$$m^{6}-4m^{5}+4\left(m^{2}\right)^{2}+2\left(m^{4}-2m^{3}\right)+m^{2}$$

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

$$m^{6}-4m^{5}+4m^{4}+2\left(m^{4}-2m^{3}\right)+m^{2}$$

Use the distributive property to multiply 2 by m^{4}-2m^{3}.

$$m^{6}-4m^{5}+4m^{4}+2m^{4}-4m^{3}+m^{2}$$

Combine 4m^{4} and 2m^{4} to get 6m^{4}.

$$m^{6}-4m^{5}+6m^{4}-4m^{3}+m^{2}$$

Expand

$$m^{6}-4m^{5}+6m^{4}-4m^{3}+m^{2}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(m^{3}-2m^{2}\right)^{2}.

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

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

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

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

$$m^{6}-4m^{5}+4\left(m^{2}\right)^{2}+2\left(m^{4}-2m^{3}\right)+m^{2}$$

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

$$m^{6}-4m^{5}+4m^{4}+2\left(m^{4}-2m^{3}\right)+m^{2}$$

Use the distributive property to multiply 2 by m^{4}-2m^{3}.

$$m^{6}-4m^{5}+4m^{4}+2m^{4}-4m^{3}+m^{2}$$

Combine 4m^{4} and 2m^{4} to get 6m^{4}.

$$m^{6}-4m^{5}+6m^{4}-4m^{3}+m^{2}$$