Solve the following equations and simplify the expressions: 1. 2x2 + 4x + 3 = 0 2. x2 + 3 3. x2 - (1+3) + 4x

here are the solutions and simplifications for the expressions provided:

• Equation 1: $2x^2 + 4x + 3 = 0$

  • Step 1: Identify coefficients. This is a quadratic equation in the form $ax^2 + bx + c = 0$, where $a=2$, $b=4$, and $c=3$.

  • Step 2: Calculate the discriminant. The discriminant is $\Delta = b^2 - 4ac$.

    $$\Delta = (4)^2 - 4(2)(3)$$ $$\Delta = 16 - 24$$ $$\Delta = -8$$

  • Step 3: Find the solutions using the quadratic formula. Since the discriminant is negative, there are no real solutions. The solutions are complex conjugates.

    $$x = \frac{-b \pm \sqrt{\Delta}}{2a}$$ $$x = \frac{-4 \pm \sqrt{-8}}{2(2)}$$ $$x = \frac{-4 \pm \sqrt{8}i}{4}$$ $$x = \frac{-4 \pm 2\sqrt{2}i}{4}$$ $$x = \frac{-2 \pm \sqrt{2}i}{2}$$ $$x = -1 \pm \frac{\sqrt{2}}{2}i$$

  • Final Answer for Equation 1: The solutions are $\boxed{x = -1 + \frac{\sqrt{2}}{2}i, x = -1 - \frac{\sqrt{2}}{2}i}$.

• Expression 2: $(x^2 + 3)$

  • Step 1: Simplify the expression. This expression is already in its simplest form.

  • Final Answer for Expression 2: The simplified expression is $\boxed{x^2 + 3}$.

• Expression 3: $x^2 - (1+3) + 4x$

  • Step 1: Evaluate the term inside the parenthesis.

    $$1+3 = 4$$

  • Step 2: Substitute the value back into the expression.

    $$x^2 - 4 + 4x$$

  • Step 3: Rearrange the terms in standard form (descending powers of $x$).

    $$x^2 + 4x - 4$$

  • Final Answer for Expression 3: The simplified expression is $\boxed{x^2 + 4x - 4}$.