Factorize the Expression: Breaking Down 15a²+10a+5

Factor the given expression:

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

We must refer to the numbers and letters separately, remembering that a common factor is a factor (multiplier) common to all the 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, hence 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 can summarise as follows:

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

$5\cdot1\\ \downarrow\\ 5$Let's take the above value as a multiple outside the parenthesis and ask the question: "How many times must we multiply the common factor (including its sign) in order to obtain the terms of the original expression (including their signs)?" Using this method we can determine 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 refer both to the sign of the common factor (in black) that we extracted as a multiple outside of the parenthesis, as well as to the sign of the terms in the original expression (in colors) Note that there is no obligation to show it. Whilst the above method is described in stages it is recommended to jump directly to the broken down form in the last line whilst continuing to refer to the previous signs which are an integral part of the expression.

We can easily ensure that this decomposition is correct by opening the parentheses with the help of the distributive property. As such we can ensure that the original expression that we decomposed can be effectively recovered - Remember, this must be done emphasizing the sign of the members in the original expression as well as the sign (which is always selectable) of the common factor.

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

Therefore, the correct answer is option b.