Question 1209335
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First we have to determine a few terms of {{{f^2}}} i.e. {{{( f(x) )^2}}}


I'll use the <a href="https://www.algebra.com/algebra/homework/playground/lessons/box-method.lesson">box method</a>.


{{{( f(x) )^2}}} is the same as {{{ f(x) * f(x) }}}
Write the terms of f(x) along the top row and along the left hand side
<table border = "1" cellpadding = "5"><tr><td></td><td>x^3</td><td>3x^2</td><td>4x</td><td>-7</td></tr><tr><td>x^3</td><td></td><td></td><td></td><td></td></tr><tr><td>3x^2</td><td></td><td></td><td></td><td></td></tr><tr><td>4x</td><td></td><td></td><td></td><td></td></tr><tr><td>-7</td><td></td><td></td><td></td><td></td></tr></table>
We then fill out this table by multiplying the headers (eg: 7 times 7 = 49 in the bottom right corner)


We don't have to fill out the entire table. 
Since we only want the coefficient x^2 at the end, we just need the terms that have exponent 2 or smaller.
<table border = "1" cellpadding = "5"><tr><td></td><td>x^3</td><td>3x^2</td><td>4x</td><td>-7</td></tr><tr><td>x^3</td><td></td><td></td><td></td><td></td></tr><tr><td>3x^2</td><td></td><td></td><td></td><td>-21x^2</td></tr><tr><td>4x</td><td></td><td></td><td>16x^2</td><td>-28x</td></tr><tr><td>-7</td><td></td><td>-21x^2</td><td>-28x</td><td>49</td></tr></table>
Combining like terms gives -26x^2-56x+49


This demonstrates that {{{ ( f(x) )^2 = h(x) -26x^2-56x+49}}} where h(x) is some 6th degree polynomial and its last term is some cubic monomial. 
We don't need to worry about h(x) since it won't influence the x^2 coefficient at the end.


We'll use the box method again.
Since h(x) doesn't play a role, we can simply ignore it to focus on the -26x^2-56x+49 portion.
Write those terms along the top row and the terms of g(x) along the left column. 
<table border = "1" cellpadding = "5"><tr><td></td><td>-26x^2</td><td>-56x</td><td>49</td></tr><tr><td>2x^4</td><td></td><td></td><td></td></tr><tr><td>-8x^3</td><td></td><td></td><td></td></tr><tr><td>4x^2</td><td></td><td></td><td>196x^2</td></tr><tr><td>-1</td><td>26x^2</td><td></td><td></td></tr></table>
Any cell that's blank won't play a role in the final answer.
The terms that do contribute a role are the 26x^2 and 196x^2. 
Those add to <font color=red>222</font>x^2



If you wanted, you can expand out every term of {{{f(x)*f(x)*g(x)}}} to get 
{{{2x^10 + 4x^9 - 10x^8 - 92x^7 - 65x^6 + 130x^5 + 425x^4 - 626x^3 + red(222)x^2 + 56x - 49}}} 
but it would take a bit longer than the previous method discussed above.
Or you can use an online calculator such as GeoGebra, WolframAlpha, etc to quickly arrive at that massive 10th degree polynomial. 
I recommend to use such a specialized calculator only if you are checking your answer. 



Answer: <font color=red>222</font>
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