Solve (m+2n)(p+q)=m(p+q)+2n(p+q)

$$( m + 2 n ) ( p + q ) = m ( p + q ) + 2 n ( p + q )$$

Solve for m (complex solution)

$$m\in \mathrm{C}$$

Solution Steps

Use the distributive property to multiply m+2n by p+q.

$$mp+mq+2np+2nq=m\left(p+q\right)+2n\left(p+q\right)$$

Use the distributive property to multiply m by p+q.

$$mp+mq+2np+2nq=mp+mq+2n\left(p+q\right)$$

Use the distributive property to multiply 2n by p+q.

$$mp+mq+2np+2nq=mp+mq+2np+2nq$$

Subtract mp from both sides.

$$mp+mq+2np+2nq-mp=mq+2np+2nq$$

Combine mp and -mp to get 0.

$$mq+2np+2nq=mq+2np+2nq$$

Subtract mq from both sides.

$$mq+2np+2nq-mq=2np+2nq$$

Combine mq and -mq to get 0.

$$2np+2nq=2np+2nq$$

Reorder the terms.

$$\text{true}$$

This is true for any m.

$$m\in \mathrm{C}$$

Solve for n (complex solution)

$$n\in \mathrm{C}$$

Solution Steps

Use the distributive property to multiply m+2n by p+q.

$$mp+mq+2np+2nq=m\left(p+q\right)+2n\left(p+q\right)$$

Use the distributive property to multiply m by p+q.

$$mp+mq+2np+2nq=mp+mq+2n\left(p+q\right)$$

Use the distributive property to multiply 2n by p+q.

$$mp+mq+2np+2nq=mp+mq+2np+2nq$$

Subtract 2np from both sides.

$$mp+mq+2np+2nq-2np=mp+mq+2nq$$

Combine 2np and -2np to get 0.

$$mp+mq+2nq=mp+mq+2nq$$

Subtract 2nq from both sides.

$$mp+mq+2nq-2nq=mp+mq$$

Combine 2nq and -2nq to get 0.

$$mp+mq=mp+mq$$

Reorder the terms.

$$\text{true}$$

This is true for any n.

$$n\in \mathrm{C}$$

Solve for m

$$m\in \mathrm{R}$$

Solution Steps

Use the distributive property to multiply m+2n by p+q.

$$mp+mq+2np+2nq=m\left(p+q\right)+2n\left(p+q\right)$$

Use the distributive property to multiply m by p+q.

$$mp+mq+2np+2nq=mp+mq+2n\left(p+q\right)$$

Use the distributive property to multiply 2n by p+q.

$$mp+mq+2np+2nq=mp+mq+2np+2nq$$

Subtract mp from both sides.

$$mp+mq+2np+2nq-mp=mq+2np+2nq$$

Combine mp and -mp to get 0.

$$mq+2np+2nq=mq+2np+2nq$$

Subtract mq from both sides.

$$mq+2np+2nq-mq=2np+2nq$$

Combine mq and -mq to get 0.

$$2np+2nq=2np+2nq$$

Reorder the terms.

$$\text{true}$$

This is true for any m.

$$m\in \mathrm{R}$$

Solve for n

$$n\in \mathrm{R}$$

Solution Steps

Use the distributive property to multiply m+2n by p+q.

$$mp+mq+2np+2nq=m\left(p+q\right)+2n\left(p+q\right)$$

Use the distributive property to multiply m by p+q.

$$mp+mq+2np+2nq=mp+mq+2n\left(p+q\right)$$

Use the distributive property to multiply 2n by p+q.

$$mp+mq+2np+2nq=mp+mq+2np+2nq$$

Subtract 2np from both sides.

$$mp+mq+2np+2nq-2np=mp+mq+2nq$$

Combine 2np and -2np to get 0.

$$mp+mq+2nq=mp+mq+2nq$$

Subtract 2nq from both sides.

$$mp+mq+2nq-2nq=mp+mq$$

Combine 2nq and -2nq to get 0.

$$mp+mq=mp+mq$$

Reorder the terms.

$$\text{true}$$

This is true for any n.

$$n\in \mathrm{R}$$