Question 197424
I have tried everything for this problem and I cannot get the answer.
The directions say to expand it.
<pre><font size = 4 color= "indigo"><b> 
I have tried everything for this problem and I cannot get the answer.
The directions say to expand it.

 (t+2)(t<sup>2</sup>+4t-3)

There are 2 terms in the left set of parentheses,
and 3 terms in the right set of parentheses.  Therefore
there are 2 x 3 or 6 multiplications we must make to
remove both parentheses:

Multiply the <font color = "red">t</font> in the left set of parentheses by the
the <font color = "red">t<sup>2</sup></font> in the right set of parentheses, getting
t<sup>3</sup>, so we write that down on the line below it:

 (<font color = "red">t</font>+2)(<font color = "red">t<sup>2</sup></font>+4t-3)

  t<sup>3</sup>

Multiply the <font color = "red">t</font> in the left set of parentheses by the
the <font color = "red">+4t</font> in the right set of parentheses, getting
+4t<sup>2</sup>, so we write that down next:

 (<font color = "red">t</font>+2)(t<sup>2</sup><font color = "red">+4t</font>-3)

  t<sup>3</sup>+4t<sup>2</sup>

Multiply the <font color = "red">t</font> in the left set of parentheses by the
the <font color = "red">-3</font> in the right set of parentheses, getting
-3t, so we write that down next:  

 (<font color = "red">t</font>+2)(t<sup>2</sup>+4t<font color = "red">-3</font>)

  t<sup>3</sup>+4t<sup>2</sup>-3t

---

Multiply the <font color = "red">+2</font> in the left set of parentheses by the
the <font color = "red">t<sup>2</sup></font> in the right set of parentheses, getting
+2t<sup>2</sup>, so we write that:

 (t<font color = "red">+2</font>)(<font color = "red">t<sup>2</sup></font>+4t-3)

  t<sup>3</sup>+4t<sup>2</sup>-3t+2t<sup>2</sup> 

Multiply the <font color = "red">+2</font> in the left set of parentheses by the
the <font color = "red">+4t</font> in the right set of parentheses, getting
+8t, so we write that down next:

 (t<font color = "red">+2</font>)(t<sup>2</sup><font color = "red">+4t</font>-3)

  t<sup>3</sup>+4t<sup>2</sup>-3t+2t<sup>2</sup>+8t

Multiply the <font color = "red">+2</font> in the left set of parentheses by the
the <font color = "red">-3</font> in the right set of parentheses, getting
<font color = "red">-6</font>, so we write that down next:  

 (t<font color = "red">+2</font>)(t<sup>2</sup>+4t<font color = "red">-3</font>)

  t<sup>3</sup>+4t<sup>2</sup>-3t+2t<sup>2</sup>+8t-6

At this stage, before combining like terms, we should 
always count the terms to make sure we have the same
number of terms as the number of terms in the set of 
parentheses on the left times the number of terms in 
the set of parentheses on the right, which in this
case is 2x3 or 6. And we do have 6 terms before we
combine any of them.

Finally we combine all like terms:

  t<sup>3</sup>+4t<sup>2</sup>-3t+2t<sup>2</sup>+8t-6

We combine the +4t<sup>2</sup> and the +2t<sup>2</sup>
to get +6t<sup>2</sup>.  Also we combine the -3t 
and the +8t to get +t.  So we end up with:

  t<sup>3</sup>+6t<sup>2</sup>+5t-6

That's it!  Notice how the red expressions below form 
a pattern of the 6 multiplications of every term in the 
left set of parentheses times every term in the right 
set of parentheses:

1. (<font color = "red">t</font>+2)(<font color = "red">t<sup>2</sup></font>+4t-3)
2. (<font color = "red">t</font>+2)(t<sup>2</sup><font color = "red">+4t</font>-3)
3. (<font color = "red">t</font>+2)(t<sup>2</sup>+4t<font color = "red">-3</font>)
4. (t<font color = "red">+2</font>)(<font color = "red">t<sup>2</sup></font>+4t-3)
5. (t<font color = "red">+2</font>)(t<sup>2</sup><font color = "red">+4t</font>-3)
6. (t<font color = "red">+2</font>)(t<sup>2</sup>+4t<font color = "red">-3</font>)

Edwin</pre>