How do you factor the expression $12{{x}^{2}}+60x+75$?

Hint: The above given question $12{{x}^{2}}+60x+75$ is a quadratic equation. A quadratic equation is any equation that can be arranged in standard form as $a{{x}^{2}}+bx+c=0$ where x represents an unknown, and a, b, and c represent constant and a must be non zero constant. If a is equal to zero then the equation is linear, not quadratic. We have to factorize the above given question $12{{x}^{2}}+60x+75$. Factorization consists of writing a number or another mathematical object as a product of several factors, simpler objects of the same kind.

Complete step by step solution:
The given equation is:
$\Rightarrow 12{{x}^{2}}+60x+75$
Here we can easily see that the whole expression is easily divisible by $3$, so we will separate out that $3$ from the above equation as common factor, then we get,
$\Rightarrow 3\left( 4{{x}^{2}}+20x+25 \right)$
Now in the above there is no common factor. Now we can write $4{{x}^{2}}={{\left( 2x \right)}^{2}}$ and $25={{\left( 5 \right)}^{2}}$ both are perfect squares and we can write $20x=\left( 2x \right)\left( 5 \right)\left( 2 \right)$ , it means we can rewrite the above equation by using the formula ${{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab$, then we get,
$\begin{align}
  & \Rightarrow 3\left( {{\left( 2x \right)}^{2}}+{{\left( 5 \right)}^{2}}+2\left( 2x \right)\left( 5 \right) \right) \\
 & \Rightarrow 3{{\left( 2x+5 \right)}^{2}} \\
\end{align}$
Hence we get the factor of the above given equation $12{{x}^{2}}+60x+75$ is $3{{\left( 2x+5 \right)}^{2}}$.

Note: We can also check whether our above factor form is correct or not. We will simply open the above factor forms by using the formula ${{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab$ where a is equal to 2x and b is equal to 5, now by solving then we get,
$\begin{align}
  & \Rightarrow {{\left( 2x+5 \right)}^{2}}=4{{x}^{2}}+25+2x\times 2\times 5 \\
 & \Rightarrow {{\left( 2x+5 \right)}^{2}}=4{{x}^{2}}+25+20x \\
\end{align}$
Now multiply the whole above equation by 3 then we get,
$\begin{align}
  & \Rightarrow 3{{\left( 2x+5 \right)}^{2}}=3\left( 4{{x}^{2}}+25+20x \right) \\
 & \Rightarrow 3{{\left( 2x+5 \right)}^{2}}=12{{x}^{2}}+75+60x \\
\end{align}$
Hence we get that our factor of the given equation $12{{x}^{2}}+60x+75$ is correct which is $3{{\left( 2x+5 \right)}^{2}}$.