1. Problem 1: One complex solution to $f(x)=x^2+2x+5$ is $w_1=-1+2i$ and we must determine the other solution $w_2$ and state the relationship between $w_1$ and $w_2$.\n2. Formula: For a quadratic $ax^2+bx+c=0$ the roots are $$w=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$.\n3. Important rule: If the coefficients $a,b,c$ are real then any nonreal complex roots occur in conjugate pairs.\n4. Apply with $a=1$, $b=2$, $c=5$ and compute the discriminant $b^2-4ac=4-20=-16$.\n5. Compute the roots: $$w=\frac{-2\pm\sqrt{-16}}{2}=\frac{-2\pm4i}{2}=-1\pm2i$$.\n6. Hence the other solution is $w_2=-1-2i$ and $w_1$ and $w_2$ are complex conjugates.\n7. Problem 2: Determine the modulus of $w_1=-1+2i$.\n8. Definition: The modulus of $a+bi$ is $|a+bi|=\sqrt{a^2+b^2}$.\n9. Compute: $|w_1|=\sqrt{(-1)^2+2^2}=\sqrt{1+4}=\sqrt{5}$.\n10. Problem 3: Solve the inequality $|3x+2|\le 5$ and determine the range of $x$.\n11. Rule: $|A|\le B$ with $B\ge0$ is equivalent to $-B\le A\le B$.\n12. Apply: $$-5\le 3x+2\le 5$$.\n13. Subtract 2: $$-7\le 3x\le 3$$.\n14. Divide by 3: $$-\tfrac{7}{3}\le x\le 1$$.\n15. Therefore the solution set is $x\in[-\tfrac{7}{3},\,1]$.\n16. Problem 4: Find the coefficient of the $x$ term in the expansion of $\left(5-\tfrac{x}{3}\right)^3$.\n17. Formula: $(a+b)^3=a^3+3a^2b+3ab^2+b^3$.\n18. Here $a=5$ and $b=-\tfrac{x}{3}$, so the linear-in-$x$ term comes from $3a^2b$.\n19. Compute: $3a^2b=3\cdot25\cdot\left(-\tfrac{x}{3}\right)=25\cdot(-x)=-25x$.\n20. Hence the coefficient of $x$ is $-25$.\n21. Problem 5: Using the purchases, write the two linear equations corresponding to the other purchases given prices $x$ (pencil), $y$ (pen), $z$ (eraser).\n22. From the data the second purchase (one month later) is 3 pencils, 1 pen and 1 eraser for 33, giving $3x+y+z=33$.\n23. The third purchase is 1 pencil and 4 pens for 45, giving $x+4y=45$.\n24. Problem 6: Construct the augmented matrix $[A|b]$ for the system using the three equations $4x+3y+3z=74$, $3x+y+z=33$, $x+4y=45$.\n25. Coefficients: the third equation has coefficient 0 for $z$.\n26. The augmented matrix is $$[A|b]=\begin{bmatrix}4 & 3 & 3 & 74\\ 3 & 1 & 1 & 33\\ 1 & 4 & 0 & 45\end{bmatrix}$$.\n27. Problem 7: Solve $\dfrac{9^{2x+1}}{3^x}=1$.\n28. Rewrite $9=3^2$ so $9^{2x+1}=3^{4x+2}$.\n29. Then $$\frac{3^{4x+2}}{3^x}=3^{3x+2}=1$$.\n30. Since $3^t=1\iff t=0$, set $3x+2=0$ giving $x=-\tfrac{2}{3}$.\n31. Problem 8: Solve $\log(3x+1)=2$ (common logarithm presumed).\n32. Convert to exponential form: $3x+1=10^2=100$.\n33. Then $3x=99$ and $x=33$.\n34. Problem 9: Prove that for $f(x)=\ln\!\left(\dfrac{1+x}{1-x}\right)$ with $-1