1. **Problem statement:** b) Find the expression for $M$ such that $$M + (5x^2 - 2xy) = 6x^2 + 9xy - y^2.$$ c) Find $x$ such that $$ (8x + 2)(1 - 3x) + (6x - 1)(4x - 10) = -50.$$ --- 2. **Solving part b:** We want to isolate $M$ from the equation: $$M + (5x^2 - 2xy) = 6x^2 + 9xy - y^2.$$ Subtract $(5x^2 - 2xy)$ from both sides: $$M = 6x^2 + 9xy - y^2 - (5x^2 - 2xy).$$ 3. **Simplify the right side:** $$M = 6x^2 + 9xy - y^2 - 5x^2 + 2xy.$$ Group like terms: $$M = (6x^2 - 5x^2) + (9xy + 2xy) - y^2 = x^2 + 11xy - y^2.$$ --- 4. **Solving part c:** Given: $$(8x + 2)(1 - 3x) + (6x - 1)(4x - 10) = -50.$$ 5. **Expand each product:** $$(8x + 2)(1 - 3x) = 8x \cdot 1 - 8x \cdot 3x + 2 \cdot 1 - 2 \cdot 3x = 8x - 24x^2 + 2 - 6x = -24x^2 + (8x - 6x) + 2 = -24x^2 + 2x + 2.$$ $$(6x - 1)(4x - 10) = 6x \cdot 4x - 6x \cdot 10 - 1 \cdot 4x + 1 \cdot 10 = 24x^2 - 60x - 4x + 10 = 24x^2 - 64x + 10.$$ 6. **Sum the two expressions:** $$(-24x^2 + 2x + 2) + (24x^2 - 64x + 10) = (-24x^2 + 24x^2) + (2x - 64x) + (2 + 10) = 0 - 62x + 12.$$ 7. **Set equal to -50:** $$-62x + 12 = -50.$$ 8. **Solve for $x$:** Subtract 12 from both sides: $$-62x + \cancel{12} - \cancel{12} = -50 - 12,$$ $$-62x = -62.$$ Divide both sides by $-62$: $$\frac{\cancel{-62}x}{\cancel{-62}} = \frac{-62}{-62},$$ $$x = 1.$$ --- **Final answers:** b) $$M = x^2 + 11xy - y^2.$$ c) $$x = 1.$$