d) \( \frac{5 x+2}{5}-\frac{3 x+10}{3} \quad \) f) \( \frac{2 x+3}{x}+\frac{7}{4 x^{2}}-\frac{4 x+1}{3} \)

We will simplify each expression step‐by‐step. ────────────────────────────── Problem (d):  Simplify (5x + 2)/5 – (3x + 10)/3 Step 1. Find a common denominator for 5 and 3. The least common denominator is 15. Step 2. Rewrite each fraction with denominator 15:  • Multiply (5x + 2)/5 by 3/3:    (5x + 2)/5 = 3(5x + 2) / (3·5) = (15x + 6)/15.  • Multiply (3x + 10)/3 by 5/5:    (3x + 10)/3 = 5(3x + 10) / (5·3) = (15x + 50)/15. Step 3. Now combine the fractions:   (15x + 6)/15 – (15x + 50)/15    = [ (15x + 6) – (15x + 50) ] / 15 Step 4. Simplify the numerator:   (15x + 6) – (15x + 50) = 15x – 15x + 6 – 50 = –44 Thus, the simplified expression is:   –44/15 ────────────────────────────── Problem (f):  Simplify (2x + 3)/x + 7/(4x²) – (4x + 1)/3 Step 1. Identify the denominators: x, 4x², and 3.  The least common denominator (LCD) is 12x² (since LCD of x, 4x², and 3 is LCM( x, 4x², 3 ) = 12x²). Step 2. Express each fraction with the denominator 12x²: • For (2x + 3)/x:  Multiply numerator and denominator by 12x:   (2x + 3)/x = [12x(2x + 3)]/(12x²) = (24x² + 36x)/(12x²). • For 7/(4x²):  Multiply numerator and denominator by 3:   7/(4x²) = (7·3)/(4x²·3) = 21/(12x²). • For (4x + 1)/3:  Multiply numerator and denominator by 4x²:   (4x + 1)/3 = [4x²(4x + 1)]/(3·4x²) = (16x³ + 4x²)/(12x²).  Remember the minus sign in front of this term. Step 3. Now combine all fractions:   [ (24x² + 36x) + 21 – (16x³ + 4x²) ] / (12x²) Step 4. Simplify the numerator by combining like terms:   –16x³ + (24x² – 4x²) + 36x + 21    = –16x³ + 20x² + 36x + 21 Thus, the simplified expression is:   (–16x³ + 20x² + 36x + 21) / (12x²) ────────────────────────────── Final Answers: (d) (5x + 2)/5 – (3x + 10)/3 = –44/15 (f) (2x + 3)/x + 7/(4x²) – (4x + 1)/3 = (–16x³ + 20x² + 36x + 21)/(12x²)