# How do you simplify (4p^2q + 5p^3q^2 - 6pq) - ( -2p^3q^2 + 5pq - 8p^2q)?

Apr 11, 2016

$\text{ } 7 {p}^{3} {q}^{2} + 12 {p}^{2} q - 11 p q$

Basically this is an exercise for keeping track.

#### Explanation:

Example: Suppose we had $- \left(6 a - 4 b\right)$

This can be written as:$\text{ } \left(- 1\right) \times \left(6 a - 4 b\right)$

Multiply everything inside the brackets by $\left(- 1\right)$ giving:
$- 6 a + 4 b$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Applying the same principle

Consider the right hand side expression:

We have: $\left(- 1\right) \times \left(- 2 {p}^{3} {q}^{2} + 5 p q - 8 {p}^{2} q\right)$

$\implies + 2 {p}^{3} {q}^{2} - 5 p q + 8 {p}^{2} q$

Putting it all together

$\text{ } 4 {p}^{2} q + 5 {p}^{3} {q}^{2} - 6 p q + 2 {p}^{3} {q}^{2} - 5 p q + 8 {p}^{2} q$

Grouping terms

$\text{ } \left(4 {p}^{2} q + 8 {p}^{2} q\right) + \left(5 {p}^{3} {q}^{2} + 2 {p}^{3} {q}^{2}\right) + \left(- 6 p q - 5 p q\right)$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Think of $+$ as 'put with'. That way we can write

$+ \left(- 6 p q - 5 p q\right)$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\text{ } 12 {p}^{2} q + 7 {p}^{3} {q}^{2} - 11 p q$

Changing the order based on $p$

$\text{ } 7 {p}^{3} {q}^{2} + 12 {p}^{2} q - 11 p q$