I am stuck on this one question where I have to find the coefficient of x^2. It is non-calculator.

Question

What is the coefficient of $x^2$ in $$ \left(4-x^2\right)\left[\left(1+2x+3x^2\right)^6-\left(1+4x^3\right)^5\right] $$

A calculator cannot be used.

Answer 1

No amount of looking at this solution will produce enough facility for you to do these problems yourself. You should make yourself understand the idea of this answer -- that you can ignore anything with degree $>2$ -- and then try to do the problem using that idea.

Since we only want the $x^2$ coefficient, we don't care about any power of $x$ greater than $2$ at any point in the computation. So from $$ \left(4-x^2\right)\left[\left(1+2x+3x^2\right)^6-\left(1+4x^3\right)\right] $$ we can immediately restrict to $$ \left(4-x^2\right)\left[\left(1+2x+3x^2\right)^6-\left(1\right)\right] $$ Then \begin{align*} (1+2x+3x^2)^6 &= ((1+2x+3x^2)^2)^3 \\ &= \left( (1+2x+3x^2) \cdot 1 + (1+2x+3x^2)\cdot 2x + {} \right. \\ &\qquad \left. (1+2x+3x^2) \cdot 3x^2) \right)^3 \\ &= \left((1+2x+3x^2) + {} \right. \\ &\qquad (1 + 2x + \text{[don't care]})\cdot 2x + {} \\ &\qquad \left. (1 + \text{[don't care]})\cdot 3x^2 \right)^3 \\ &= (1+4x+10x^2)^3 \\ &= (1+4x+10x^2) \cdot (1+4x+10x^2) \cdot (1+4x+10x^2) \\ &= (1 + 4x + 4x + 10x^2 + 16 x^2 + 10x^2 + \text{[don't care]}) \cdot (1+4x+10x^2) \\ &= (1 + 8x + 36 x^2 + \text{[don't care]}) \cdot (1+4x+10x^2) \\ &= 1 + 4x + 8x + 36 x^2 + 32 x^2 + 10x^2 + \text{[don't care]} \\ &= 1 + 12 x + 78 x^2 + \text{[don't care]} \text{.} \end{align*} There are many ways to keep track of the bookkeeping in the above. It can even be done entirely mentally, with practice. Now we can reduce our interest to $$ \left(4-x^2\right)\left[12 x + 78 x^2 \right] = 48x + 312 x^2 + \text{[don't care]} $$ and we are done.

Answer 2

HINT

Let $\left\{x^2\right\} A(x)$ denote the coefficient of $x^2$ in the expression $A(x)$. Then, $$ \begin{split} \left\{x^2\right\} \left(4-x^2\right)[f(x)-g(x)] &= 4\left\{x^2\right\}[f(x)-g(x)] - \left\{x^0\right\}[f(x)-g(x)] \\ \end{split} $$ So what are the coefficients of $x^0$ and $x^2$ in your expressions?