How do you solve $3\left( 3u+2 \right)+5=2\left( 2u-2 \right)$ ?

Hint: At first, we apply distributive property to the $3\left( 3u+2 \right)$ term and then to the $2\left( 2u-2 \right)$ term and break it down by opening the bracket. We now get algebraic and arithmetic terms. After that, we take the algebraic terms to one side and the arithmetic terms to the other side. We then go on simplifying the equation by the arithmetic operations as demanded.

Complete step by step solution:
The given equation that we have is,
$3\left( 3u+2 \right)+5=2\left( 2u-2 \right)$
We start off the solution by applying the distributive property to the first term of the left hand side of the equation. The distributive property states that an expression of the form $a\left( b+c \right)$ can be written as $ab+ac$ . By comparing the first term of the left hand side of the above equation with the general form, we get,
$a=3,b=3u,c=2$
Thus, applying the distributive property, it becomes, $3\left( 3u+2 \right)=3\times 3u+3\times 2$ which upon simplification gives, $9u+6$ . We then apply the distributive property to the first term of the right hand side of the equation. By comparing the first term of the right hand side of the above equation with the general form, we get,
$a=2,b=2u,c=-2$
Thus, applying the distributive property, it becomes, $2\left( 2u-2 \right)=2\times 2u-2\times 2$ which upon simplification gives, $4u-4$ . Thus, the equation can be rewritten as,
$\Rightarrow 9u+6+5=4u-4$
Adding the $6$ and $5$ , we get,
$\Rightarrow 9u+11=4u-4$
Subtracting $11$ from both sides of the above equation, we get,
$\Rightarrow 9u+11-11=4u-4-11$
This upon simplification gives,
$\Rightarrow 9u=4u-15$
Subtracting $4u$ from both sides of the above equation, we get,
$\Rightarrow 9u-4u=4u-4u-15$
This upon simplification gives,
$\Rightarrow 5u=-15$
Dividing both sides of the above equation by $5$ , we get,
$\Rightarrow \dfrac{5u}{5}=\dfrac{-15}{5}$
This upon simplification gives,
$\Rightarrow u=-3$
Therefore, we can conclude that the solution of the given equation is $u=-3$ .

Note: In this problem, we must be careful while applying the distributive property as students often overlook the negative terms inside the brackets and end up in wrong answers. We should finally cross check our answer by putting the value of the variable in the equation. The BODMAS rule must be strictly followed.