1. **State the problem:** Simplify the expression $$-3 \cdot (-2x^2 + 3x - 1) + 2 \cdot (4x^2 - 5x + 3) - \frac{5}{8} \cdot (-3x^2 - 2x + 7)$$. 2. **Recall the distributive property:** For any numbers $a$, $b$, and $c$, $$a(b + c) = ab + ac$$. We will apply this to each term. 3. **Distribute each multiplication:** - $$-3 \cdot (-2x^2 + 3x - 1) = (-3)(-2x^2) + (-3)(3x) + (-3)(-1) = 6x^2 - 9x + 3$$ - $$2 \cdot (4x^2 - 5x + 3) = 2(4x^2) + 2(-5x) + 2(3) = 8x^2 - 10x + 6$$ - $$- \frac{5}{8} \cdot (-3x^2 - 2x + 7) = - \frac{5}{8}(-3x^2) - \frac{5}{8}(-2x) - \frac{5}{8}(7) = \frac{15}{8}x^2 + \frac{10}{8}x - \frac{35}{8}$$ 4. **Combine all results:** $$6x^2 - 9x + 3 + 8x^2 - 10x + 6 + \frac{15}{8}x^2 + \frac{10}{8}x - \frac{35}{8}$$ 5. **Group like terms:** - For $x^2$: $$6x^2 + 8x^2 + \frac{15}{8}x^2 = \left(6 + 8 + \frac{15}{8}\right)x^2 = \left(14 + \frac{15}{8}\right)x^2 = \frac{112}{8}x^2 + \frac{15}{8}x^2 = \frac{127}{8}x^2$$ - For $x$: $$-9x - 10x + \frac{10}{8}x = (-19 + \frac{10}{8})x = \left(-19 + 1.25\right)x = -17.75x = -\frac{71}{4}x$$ - Constants: $$3 + 6 - \frac{35}{8} = 9 - \frac{35}{8} = \frac{72}{8} - \frac{35}{8} = \frac{37}{8}$$ 6. **Final simplified expression:** $$\frac{127}{8}x^2 - \frac{71}{4}x + \frac{37}{8}$$ This is the simplified form of the original expression.