b We are given the following equation with variable

m .

(m + 2) (m + 3) = (m + 2) (m - 2)

Note that the left-hand side and right-hand side are both products where

(m + 2)

is a factor on both sides. By dividing by

(m + 2),

we can cancel out this factor.

\frac{( m + 2 ) ( m + 3 )}{( m + 2 )} = \frac{( m + 2 ) ( m - 2 )}{( m + 2 )}

However, we have to be very careful. When we divide with something containing a variable, we automatically exclude any values of

m

that results in a division of

0 .

Therefore, we will not solve the equation this way. Let's instead multiply the parentheses in the given equation.

(m + 2) (m + 3) = (m + 2) (m - 2)

▼

Distribute

(m + 3) & (m - 2)

Distribute $(m + 3)$

m (m + 3) + 2 (m + 3) = (m + 2) (m - 2)

Distribute $m$

m^{2} + 3 m + 2 (m + 3) = (m + 2) (m - 2)

Distribute $2$

m^{2} + 3 m + 2 m + 6 = (m + 2) (m - 2)

Distribute $(m - 2)$

m^{2} + 3 m + 2 m + 6 = m (m - 2) + 2 (m - 2)

Distribute $m$

m^{2} + 3 m + 2 m + 6 = m^{2} - 2 m + 2 (m - 2)

Distribute $2$

m^{2} + 3 m + 2 m + 6 = m^{2} - 2 m + 2 m - 4

▼

Solve for

m

Add terms

m^{2} + 5 m + 6 = m^{2} - 4

$LHS - m^{2} = RHS - m^{2}$

5 m + 6 = - 4

$LHS - 6 = RHS - 6$

5 m = - 10

$LHS / 5 = RHS / 5$

m = - 2

Let's check the solution.

(m + 2) (m + 3) = (m + 2) (m - 2)

$m = - 2$

(- 2 + 2) (- 2 + 3) = ? (- 2 + 2) (- 2 - 2)

Add and subtract terms

0 (1) = ? 0 (- 4)

0 = 0 ✓

We have the correct solution.

Exercise 37 Page 156