Solve (3m-n)^2-4(2m+3n)^2

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

Evaluate

$$-\left(m+7n\right)\left(7m+5n\right)$$

Solution Steps

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

$$9m^{2}-6mn+n^{2}-4\left(2m+3n\right)^{2}$$

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

$$9m^{2}-6mn+n^{2}-4\left(4m^{2}+12mn+9n^{2}\right)$$

Use the distributive property to multiply -4 by 4m^{2}+12mn+9n^{2}.

$$9m^{2}-6mn+n^{2}-16m^{2}-48mn-36n^{2}$$

Combine 9m^{2} and -16m^{2} to get -7m^{2}.

$$-7m^{2}-6mn+n^{2}-48mn-36n^{2}$$

Combine -6mn and -48mn to get -54mn.

$$-7m^{2}-54mn+n^{2}-36n^{2}$$

Combine n^{2} and -36n^{2} to get -35n^{2}.

$$-7m^{2}-54mn-35n^{2}$$

Expand

$$-7m^{2}-54mn-35n^{2}$$

Solution Steps

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

$$9m^{2}-6mn+n^{2}-4\left(2m+3n\right)^{2}$$

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

$$9m^{2}-6mn+n^{2}-4\left(4m^{2}+12mn+9n^{2}\right)$$

Use the distributive property to multiply -4 by 4m^{2}+12mn+9n^{2}.

$$9m^{2}-6mn+n^{2}-16m^{2}-48mn-36n^{2}$$

Combine 9m^{2} and -16m^{2} to get -7m^{2}.

$$-7m^{2}-6mn+n^{2}-48mn-36n^{2}$$

Combine -6mn and -48mn to get -54mn.

$$-7m^{2}-54mn+n^{2}-36n^{2}$$

Combine n^{2} and -36n^{2} to get -35n^{2}.

$$-7m^{2}-54mn-35n^{2}$$