We need to express:
$6x^2 + 10x + 5$ in the form:
$a(x + b) + c$

Step 1: Factor Out the Leading Coefficient

The highest power of $x$ is 2, so we first factor out the coefficient of $x^2$ from the first two terms:

$6x^2 + 10x + 5 = 6(x^2 + \frac{10}{6}x) + 5$

Simplifying $\frac{10}{6}$ to $\frac{5}{3}$:

$6(x^2 + \frac{5}{3}x) + 5$

Step 2: Convert to the Desired Form

We want to express it in the form $a(x + b) + c$. Let's attempt to complete the square inside the parentheses.

1. Take half of the coefficient of $x$ inside the parentheses:
   $\frac{5}{3} \div 2 = \frac{5}{6}$
2. Square it:
   $\left(\frac{5}{6}\right)^2 = \frac{25}{36}$

Step 3: Add and Subtract Inside the Parentheses

To complete the square:

$6 \left( x^2 + \frac{5}{3}x + \frac{25}{36} - \frac{25}{36} \right) + 5$

Rewriting:

$6 \left( (x + \frac{5}{6})^2 - \frac{25}{36} \right) + 5$

Step 4: Expand and Simplify

Distribute the 6:

$6(x + \frac{5}{6})^2 - 6 \times \frac{25}{36} + 5$

$6(x + \frac{5}{6})^2 - \frac{150}{36} + 5$

$6(x + \frac{5}{6})^2 - \frac{25}{6} + 5$

Express 5 as a fraction:

$6(x + \frac{5}{6})^2 - \frac{25}{6} + \frac{30}{6}$

$6(x + \frac{5}{6})^2 + \frac{5}{6}$

Final Answer:

$a = 6, \quad b = \frac{5}{6}, \quad c = \frac{5}{6}$

Thus, the given quadratic can be rewritten as:

$6(x + \frac{5}{6})^2 + \frac{5}{6}$

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Would you like more details on any step? Here are some related questions:

1. How do you complete the square for any quadratic expression?
2. Can this form be used to find the vertex of a quadratic function?
3. How is this different from factoring a quadratic expression?
4. Can we express any quadratic expression in this form?
5. What is the significance of the constant $c$ in this form?

Tip:

When completing the square, always remember to balance the equation by subtracting and adding the same term inside the parentheses.