The problem asks to find the values of $x$ that satisfy the equation $4 + 2(5x + 2x^2) = 11$. The answer should be in the simplest surd form.

1. Problem Description

The problem asks to find the values of $x$ that satisfy the equation $4 + 2(5x + 2x^2) = 11$. The answer should be in the simplest surd form.

2. Solution Steps

First, expand the expression:

$4 + 2(5x + 2x^2) = 11$

$4 + 10x + 4x^2 = 11$

Rearrange the equation to the standard quadratic form:

$4x^2 + 10x + 4 - 11 = 0$

$4x^2 + 10x - 7 = 0$

Use the quadratic formula to solve for $x$:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a = 4$, $b = 10$, and $c = -7$.

$x = \frac{-10 \pm \sqrt{10^2 - 4(4)(-7)}}{2(4)}$

$x = \frac{-10 \pm \sqrt{100 + 112}}{8}$

$x = \frac{-10 \pm \sqrt{212}}{8}$

Simplify the square root:

$\sqrt{212} = \sqrt{4 \cdot 53} = 2\sqrt{53}$

Substitute the simplified square root back into the equation:

$x = \frac{-10 \pm 2\sqrt{53}}{8}$

Divide both the numerator and the denominator by 2:

$x = \frac{-5 \pm \sqrt{53}}{4}$

3. Final Answer

The values of $x$ are $x = \frac{-5 + \sqrt{53}}{4}$ and $x = \frac{-5 - \sqrt{53}}{4}$.