1. The problem is to transform the given quadratic equations into standard form $ax^2 + bx + c = 0$. 2. Recall that the standard form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. 3. Let's start with equation (a): $$2x^2 + (2x - 1)x - (x + 12) = 0$$ Distribute terms: $$2x^2 + 2x^2 - x - x - 12 = 0$$ Simplify: $$4x^2 - 2x - 12 = 0$$ This is the standard form for (a). 4. For equation (b): $$3x(x + 2) = 6(x - 7)$$ Distribute both sides: $$3x^2 + 6x = 6x - 42$$ Bring all terms to one side: $$3x^2 + 6x - 6x + 42 = 0$$ Simplify: $$3x^2 + 42 = 0$$ This is the standard form for (b). 5. For equation (c): $$(2 - x)^2 = 4x - (x + 1)^2$$ Expand squares: $$4 - 4x + x^2 = 4x - (x^2 + 2x + 1)$$ Simplify right side: $$4 - 4x + x^2 = 4x - x^2 - 2x - 1$$ Bring all terms to one side: $$4 - 4x + x^2 - 4x + x^2 + 2x + 1 = 0$$ Combine like terms: $$2x^2 - 6x + 5 = 0$$ This is the standard form for (c). 6. For equation (d): $$(x + 1)^2 + (x + 2)^2 = 10$$ Expand squares: $$x^2 + 2x + 1 + x^2 + 4x + 4 = 10$$ Combine like terms: $$2x^2 + 6x + 5 = 10$$ Bring all terms to one side: $$2x^2 + 6x + 5 - 10 = 0$$ Simplify: $$2x^2 + 6x - 5 = 0$$ This is the standard form for (d). Final answers: (a) $4x^2 - 2x - 12 = 0$ (b) $3x^2 + 42 = 0$ (c) $2x^2 - 6x + 5 = 0$ (d) $2x^2 + 6x - 5 = 0$