$\mathrm{with}\left(\mathrm{Student}:-\mathrm{Basics}\right)\:$

$\mathrm{LongDivision}\left(48x^4+284x^3+620x^2+593x+210\,2x+3\right)$

$\begin{array}{cc}\stackrel{\phantom{\frac{x^2}{2}}}{\mathrm{\%+}\left(2x\,3\right)}& \begin{array}{cccccc}& 24x^3& \+106x^2& \+151x& \+70& \\ \)\phantom{x^2}& \phantom{1}48x^4& \phantom{1}\+284x^3& \phantom{1}\+620x^2& \phantom{1}\+593x& \phantom{1}\+210\\ & \multicolumn{2}{c}{\frac{48x^4+72x^3}{\phantom{\.}}}& \\ & & \multicolumn{2}{c}{212x^3+620x^2}& & \\ & & \multicolumn{2}{c}{\frac{212x^3+318x^2}{\phantom{\.}}}& & \\ & & & \multicolumn{2}{c}{302x^2+593x}& & & \\ & & & \multicolumn{2}{c}{\frac{302x^2+453x}{\phantom{\.}}}& & & \\ & & & & \multicolumn{2}{c}{140x+210}& & & & \\ & & & & \multicolumn{2}{c}{\frac{140x+210}{\phantom{\.}}}& & & & \\ & & & & & 0\hfill & & & & & \end{array}\end{array}$

$\mathrm{LongDivision}\left(1001\,30\, \'\mathrm{decimaldigits}\'\=4\right)\;$

$\begin{array}{ccccccccccccccc}& & & & \underset{\—}{ }& \underset{\—}{ }& \underset{\—}{ }& \underset{\—}{ }& \underset{\—}{3}& \underset{\—}{3}& \underset{\—}{\.}& \underset{\—}{3}& \underset{\—}{6}& \underset{\—}{6}& \underset{\—}{6}\\ & 3& 0& & \)& & 1& 0& 0& 1& \.& 0& 0& 0& 0\\ & & & & & & & \underset{\—}{9}& \underset{\—}{0}\\ & & & & & & & 1& 0& 1\\ & & & & & & & & \underset{\—}{9}& \underset{\—}{0}\\ & & & & & & & & 1& 1& 0\\ & & & & & & & & & \underset{\—}{9}& \underset{\—}{0}\\ & & & & & & & & & 2& 0& 0\\ & & & & & & & & & \underset{\—}{1}& \underset{\—}{8}& \underset{\—}{0}\\ & & & & & & & & & & 2& 0& 0\\ & & & & & & & & & & \underset{\—}{1}& \underset{\—}{8}& \underset{\—}{0}\\ & & & & & & & & & & & 2& 0& 0\\ & & & & & & & & & & & \underset{\—}{1}& \underset{\—}{8}& \underset{\—}{0}\\ & & & & & & & & & & & & 2& 0\end{array}$

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{Basics}\right)\:$

$\mathrm{FactorSteps}\left(x^3+6x^2+12x+8\right)$

$\begin{array}{lll}& & \left[\right]\\ \text{▫}& & \text{1. Trial Evaluations}\\ & \text{◦}& \text{Rewrite in standard form}\\ & & \left[\right]\\ & \text{◦}& \text{The factors of the constant coefficient}8\text{are:}\\ & & C=\left\{1\,2\,4\,8\right\}\\ & \text{◦}& \text{Trial evaluations of}x\text{in}\text{±}C\text{find}x\text{=}\mathrm{-2}\text{satisfies the equation, so}x+2\text{is a factor}\\ & & \left[\right]\\ & \text{◦}& \text{Divide by}x+2\\ & & \begin{array}{cc}\stackrel{\phantom{\frac{x^2}{2}}}{\left[\right]}& \begin{array}{ccccc}& x^2& \+4x& \+4& \\ \)\phantom{x^2}& \phantom{1}{x}^3& \phantom{1}\+6x^2& \phantom{1}\+12x& \phantom{1}\+8\\ & \multicolumn{2}{c}{\frac{x^3+2x^2}{\phantom{\.}}}& \\ & & \multicolumn{2}{c}{4x^2+12x}& & \\ & & \multicolumn{2}{c}{\frac{4x^2+8x}{\phantom{\.}}}& & \\ & & & \multicolumn{2}{c}{4x+8}& & & \\ & & & \multicolumn{2}{c}{\frac{4x+8}{\phantom{\.}}}& & & \\ & & & & 0\hfill & & & & \end{array}\end{array}\\ & \text{◦}& \text{Quotient times divisor from long division}\\ & & \left[\right]\\ \text{•}& & \text{2. Examine term:}\\ & & x^2+4x+4\\ \text{▫}& & \text{3. Apply the AC Method}\\ & \text{◦}& \text{Examine quadratic}\\ & & \left(\left[\right]\right)\\ & \text{◦}& \text{Look at the coefficients,}A{x}^2+Bx+C\\ & & \left[''A''=1\,''B''=4\,''C''=4\right]\\ & \text{◦}& \text{Find factors of |AC| = |}1\cdot 4\text{| =}4\\ & & \left\{1\,2\,4\right\}\\ & \text{◦}& \text{Find pairs of the above factors, which, when multiplied equal}4\\ & & \left\{\left[\right]\,\left[\right]\right\}\\ & \text{◦}& \text{Which pairs of these factors have a}\text{sum}\text{of B =}4\text{? Found:}\\ & & \left[\right]=4\\ & \text{◦}& \text{Split the middle term to use above pair}\\ & & \left[\right]\\ & \text{◦}& \text{Factor}x\text{out of the first pair}\\ & & \left[\right]\\ & \text{◦}& \text{Factor}2\text{out of the second pair}\\ & & \left[\right]\\ & \text{◦}& x+2\text{is a common factor}\\ & & \left[\right]\\ & \text{◦}& \text{Group common factor}\\ & & \left[\right]\\ & & \text{This gives:}\\ & & \left[\right]\\ \text{•}& & \text{4. This gives:}\\ & & \left[\right]\end{array}$

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{Basics}\right)\:$

$\mathrm{SolveSteps}\left(5{\ⅇ}^{4x}=16\right)$

$\begin{array}{lll}& & \text{Let's solve}\\ & & \left[\right]=16\\ \text{â–«}& & \text{Convert from exponential equation}\\ & \text{â—¦}& \text{Divide both sides by}5\\ & & \left[\right]=\left[\right]\\ & \text{â—¦}& \text{Simplify}\\ & & \left[\right]=\frac{16}{5}\\ & \text{â—¦}& \text{Apply ln to each side}\\ & & \left[\right]=\left[\right]\\ & \text{â—¦}& \text{Apply ln rule: ln(e^b) = b}\\ & & 4x\end{array}$