Decompose the following expression into factors:

$15a^2+10a+5$

__Factor__ the given expression:

$15a^2+10a+5$We will do this by** taking out the **__greatest__ common factor, both from the numbers and the letters,

We will refer to the numbers and letters __separately__, remembering that a common factor is a factor (multiplier) **common to all terms of the expression**,

__Let's start with the numbers__

Note that the numerical coefficients of the terms in the given expression, that is, the numbers: 5,10,15 are all multiples of the number 5:

$15=3\cdot\underline{5}\\
10=2\cdot\underline{5}\\$Therefore, the number 5 **is the greatest common factor of the numbers**,

__For the letters:__

**Note that only the first two terms on the left depend on x**, the third term is a free number that does not depend on x, therefore **there is no common factor for all three terms together for the letters** (that is, we will consider the number 1 as the common factor for the letters)

__Therefore, we summarize:__

The greatest common factor __(for numbers and letters together)__ is:

$5\cdot1\\
\downarrow\\
5$Let's take it, then, as a multiple outside the parenthesis and ask the question: **"How many times will we multiply the common factor (including its sign) obtaining each of the terms of the original expression (including its sign)?",** so we will know what is the expression inside the parenthesis that multiplied the common factor:

$\textcolor{red}{ 15a^2}\textcolor{blue}{+10a} \textcolor{green}{+5} \\
\underline{5}\cdot\textcolor{red}{3a^2}+\underline{5}\cdot\textcolor{blue}{(+2a)}+\underline{5}\cdot\textcolor{green}{(+1)}\\
\downarrow\\
\underline{5}(\textcolor{red}{3a^2}\textcolor{blue}{+2a}\textcolor{green}{+1})$In the previous expression, the operation is explained through colors and signs:

The common factor has been highlighted __with an underscore__, and the multiples inside the parenthesis are associated with the terms of the original expression with the help of colors, __note that in the detail of the decomposition above we also refer to the sign of the common factor (in black) that we extracted as a multiple outside the parenthesis and the sign of the terms in the original expression (in colors),__ there is no obligation to show it. This is in stages as described above, you can (and it is worth) jump directly to the broken down form in the last line, but you definitely should refer to the previous signs, since in each member the sign is an inseparable part of it,

**We can ensure that this decomposition is correct **__easily__ by opening the parentheses __with the help of the distributive property__ and ensuring that the original expression that we decomposed is effectively obtained back - member, __this must be done emphasizing the sign of the members in the original expression and the sign (which is always selectable) of the common factor.__

(Initially, you should use the previous colors to ensure you get all the terms in the original expression and belong to the multiple inside the parenthesis; later, it is recommended not to use the colors)

__Therefore, the correct answer is option b.__