18. Which value(s) of $$ x $$ satisfies the equation below? $$ a: 5 b: 12 c:-10 $$ $$ \left(5 x^{2}+2 x\right)\left(2 x^{2}+12 x-10=0\right. $$ a. No real zeros b. $$ \{-0.7,3.1\} $$ c. $$ \{0.7,-3.1\} $$ d. $$ \{33.2,-35.6\} $$

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To solve the equation

$$
(5x^2 + 2x)(2x^2 + 12x - 10) = 0,
$$

we need to find the values of $$ x $$ that satisfy either $$ 5x^2 + 2x = 0 $$ or $$ 2x^2 + 12x - 10 = 0 $$.

### Step 1: Solve $$ 5x^2 + 2x = 0 $$

We can factor out $$ x $$:

$$
x(5x + 2) = 0.
$$

This gives us two solutions:

1. $$ x = 0 $$
2. $$ 5x + 2 = 0 $$ which simplifies to $$ x = -\frac{2}{5} = -0.4 $$.

### Step 2: Solve $$ 2x^2 + 12x - 10 = 0 $$

We will use the quadratic formula:

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

where $$ a = 2 $$, $$ b = 12 $$, and $$ c = -10 $$.

Calculating the discriminant:

$$
b^2 - 4ac = 12^2 - 4 \cdot 2 \cdot (-10) = 144 + 80 = 224.
$$

Now, substituting into the quadratic formula:

$$
x = \frac{-12 \pm \sqrt{224}}{2 \cdot 2} = \frac{-12 \pm 2\sqrt{56}}{4} = \frac{-12 \pm 2\sqrt{4 \cdot 14}}{4} = \frac{-12 \pm 4\sqrt{14}}{4} = -3 \pm \sqrt{14}.
$$

### Step 3: Calculate the approximate values

Calculating $$ \sqrt{14} $$:

$$
\sqrt{14} \approx 3.74.
$$

Thus, the two solutions are:

1. $$ x \approx -3 + 3.74 \approx 0.74 $$
2. $$ x \approx -3 - 3.74 \approx -6.74 $$

### Summary of Solutions

From both parts, we have the solutions:

1. From $$ 5x^2 + 2x = 0 $$: $$ x = 0 $$ and $$ x = -0.4 $$.
2. From $$ 2x^2 + 12x - 10 = 0 $$: $$ x \approx 0.74 $$ and $$ x \approx -6.74 $$.

### Final Values

The approximate values of $$ x $$ that satisfy the equation are:

- $$ x \approx 0 $$
- $$ x \approx -0.4 $$
- $$ x \approx 0.74 $$
- $$ x \approx -6.74 $$

Now, let's check the options provided:

- a. No real zeros
- b. $$ \{-0.7, 3.1\} $$
- c. $$ \{0.7, -3.1\} $$
- d. $$ \{33.2, -35.6\} $$

None of the options match the calculated values. Therefore, the correct answer is:

**a. No real zeros**.
The equation has no real solutions. **Simplify Answer:** No real solutions exist for the equation.

18. Which value(s) of \( x \) satisfies the equation below? \( a: 5 b: 12 c:-10 \) \[ \left(5 x^{2}+2 x\right)\left(2 x^{2}+12 x-10=0\right. \] a. No real zeros b. \( \{-0.7,3.1\} \) c. \( \{0.7,-3.1\} \) d. \( \{33.2,-35.6\} \)

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