Solve [5^2(m-n)^2]-(m+n)^2

$$[ 5 ^ { 2 } ( m - n ) ^ { 2 } ] - ( m + n ) ^ { 2 }$$

Evaluate

$$4\left(2m-3n\right)\left(3m-2n\right)$$

Solution Steps

Calculate 5 to the power of 2 and get 25.

$$25\left(m-n\right)^{2}-\left(m+n\right)^{2}$$

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

$$25\left(m^{2}-2mn+n^{2}\right)-\left(m+n\right)^{2}$$

Use the distributive property to multiply 25 by m^{2}-2mn+n^{2}.

$$25m^{2}-50mn+25n^{2}-\left(m+n\right)^{2}$$

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

$$25m^{2}-50mn+25n^{2}-\left(m^{2}+2mn+n^{2}\right)$$

To find the opposite of m^{2}+2mn+n^{2}, find the opposite of each term.

$$25m^{2}-50mn+25n^{2}-m^{2}-2mn-n^{2}$$

Combine 25m^{2} and -m^{2} to get 24m^{2}.

$$24m^{2}-50mn+25n^{2}-2mn-n^{2}$$

Combine -50mn and -2mn to get -52mn.

$$24m^{2}-52mn+25n^{2}-n^{2}$$

Combine 25n^{2} and -n^{2} to get 24n^{2}.

$$24m^{2}-52mn+24n^{2}$$

Expand

$$24m^{2}-52mn+24n^{2}$$

Solution Steps

Calculate 5 to the power of 2 and get 25.

$$25\left(m-n\right)^{2}-\left(m+n\right)^{2}$$

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

$$25\left(m^{2}-2mn+n^{2}\right)-\left(m+n\right)^{2}$$

Use the distributive property to multiply 25 by m^{2}-2mn+n^{2}.

$$25m^{2}-50mn+25n^{2}-\left(m+n\right)^{2}$$

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

$$25m^{2}-50mn+25n^{2}-\left(m^{2}+2mn+n^{2}\right)$$

To find the opposite of m^{2}+2mn+n^{2}, find the opposite of each term.

$$25m^{2}-50mn+25n^{2}-m^{2}-2mn-n^{2}$$

Combine 25m^{2} and -m^{2} to get 24m^{2}.

$$24m^{2}-50mn+25n^{2}-2mn-n^{2}$$

Combine -50mn and -2mn to get -52mn.

$$24m^{2}-52mn+25n^{2}-n^{2}$$

Combine 25n^{2} and -n^{2} to get 24n^{2}.

$$24m^{2}-52mn+24n^{2}$$