Solve the following exponential and quadratic equations algebraically for x and y: (3^x)(3^(x−2)) = 27 −2x2 + x = -3x(-2x+1) 2x + 3y + 4 = 0 and x + 2y − 5 = 0

$$3^{x + (x-2)} = 3^3$$ $$3^{2x-2} = 3^3$$

$$2x = 3 + 2$$ $$2x = 5$$ $$x = \frac{5}{2}$$

The solution is $\boxed{x = \frac{5}{2}}$.

2.3.2. Solve for $x$:

$$-2x^2 + x = -3x(-2x+1)$$

$$-2x^2 + x = (-3x)(-2x) + (-3x)(1)$$ $$-2x^2 + x = 6x^2 - 3x$$

$$0 = 6x^2 + 2x^2 - 3x - x$$ $$0 = 8x^2 - 4x$$

$$4x = 0 \quad \text{or} \quad 2x - 1 = 0$$ $$x = 0 \quad \text{or} \quad 2x = 1$$ $$x = 0 \quad \text{or} \quad x = \frac{1}{2}$$

The solutions are $\boxed{x = 0 \text{ or } x = \frac{1}{2}}$.

2.4. Solve algebraically for $x$ and $y$ in the following equations:

1. $2x + 3y + 4 = 0$
2. $x + 2y - 5 = 0$

$$-4y + 10 + 3y + 4 = 0$$ $$-y + 14 = 0$$ $$-y = -14$$ $$y = 14$$

$$x = -2(14) + 5$$ $$x = -28 + 5$$ $$x = -23$$

The solution is $\boxed{x = -23, y = 14}$.