How do you simplify #(x + 2) ^ { 2} - 5x - ( x ^ { 2} - 13)#?

1 Answer

#-x+17#

Explanation:

#(x+2)^2-5x-(x^2-13)#
^^^^^^^^^^^^^^^^^^^^^^^^^^^^
That's the whole long expression.

First, I will focus on #(x+2)^2#.
So the first thing you have to do is to apply this rule.
#(a+b)^2=a^2+2ab+b^2#

So, using that rule on #(x+2)^2#, it would be:
#x^2+2x*2+2^2#

Now we will simplify that expression. (#x^2+2x*2+2^2#)
#2*2=4#
So, #x^2+color(blue)4color(blue)x+2^2# would be the new expression.

But, #2^2=4# so the expression after simplifying would be #x^2+4x+4#.

Now, stick it into the rest of the WHOLE expression.
#x^2+4x+4-5x-(x^2-13)#
Now we will focus on the rightmost side (#-(x^2-13)#)

Distribute brackets/parentheses.
#-(x^2)-(-13)#
Now remember this: #-(a)=-a# and #-(-b)=+b#.
In this case, #a# represents #x^2# and #b# represents #13#.
That means #-(x^2)-(-13)# would become #-x^2+13#.
Stick the whole expression together and you get

#x^2+4x+4-5x-x^2+13#.

Wow. That's a long expression. Let's simplify it.
First, we need to group like terms.

#x^2+4x+4-5x-x^2+13# to
#x^2-x^2+color(blue)4color(blue)xcolor(blue)-color(blue)5color(blue)x+4+13#

You see the blue numbers/symbols?
Let's simplify those:
#4x-5x=-x#.
Now we can replace #4x-5x# with #-x#.

So, the expression would be #color(blue)x^color(blue)2color(blue)-color(blue)x^color(blue)2-x+4+13#.

Let's simplify the blue numbers/symbols again.
#x^2-x^2=0#.

The final expression would be #-x+color(blue)4color(blue)+color(blue)13#.

Add the blue numbers:
#14+3=17#

Therefore, #-x+17# is the simplified expression of
#(x+2)^2-5x-(x^2-13)#.