How do you simplify $\dfrac{4{{c}^{2}}+12c+9}{2{{c}^{2}}-11c-21}$ ?

Hint: Now to simplify the given expression we will first separate the numerator and denominator in the expression by splitting the middle term method. Now we will simplify the expression and substitute in place of numerator and denominator. Hence we have a simplified expression.

Complete step by step solution:
Now consider the given expression $\dfrac{4{{c}^{2}}+12c+9}{2{{c}^{2}}-11c-21}$
To simplify the expression we will first factorize the quadratic in numerator and denominator.
Now first consider the quadratic equation $4{{c}^{2}}+12c+9$
Now to factorize the expression we will split the middle term such that the product of the obtained terms is multiplication of first term and last term.
Hence let us write $12c=6c+6c$. Because $6c\times 6c=9{{c}^{2}}\times 4$
$\Rightarrow 9{{c}^{2}}+6c+6c+4$
Now simplifying the above equation by taking 3c common from first two terms and taking 2 common from last two terms we get,
$\begin{align}
  & \Rightarrow 3c\left( 3c+2 \right)+2\left( 3c+2 \right) \\
 & \Rightarrow \left( 3c+2 \right)\left( 3c+2 \right) \\
 & \Rightarrow {{\left( 3c+2 \right)}^{2}} \\
\end{align}$
Now consider the quadratic equation in the denominator $2{{x}^{2}}-11x-21$ .
Now again we will split the middle term in the same way.
Now we have $-11x=-14x+3x$ and $-14x\times 3x=-21\times \left( 2{{x}^{2}} \right)$
Hence splitting the middle term of the expression we get,
$\Rightarrow 2{{x}^{2}}-14x+3x-21$
Now simplifying the equation by taking 2x common from first two terms and 3 common from last two terms we get,
$\begin{align}
  & \Rightarrow 2x\left( x-7 \right)+3\left( x-7 \right) \\
 & \Rightarrow \left( 2x+3 \right)\left( x-7 \right) \\
\end{align}$
Now let us substitute the obtained values of numerator and denominator.
Hence we get, $\dfrac{{{\left( 3c+2 \right)}^{2}}}{\left( 2x+3 \right)\left( x-7 \right)}$
Hence we can write the given expression as $\dfrac{{{\left( 3c+2 \right)}^{2}}}{\left( 2x+3 \right)\left( x-7 \right)}$

Note:
Now note that to find the factors of the equation we can also use the formula to find the roots of the quadratic equation. Now the roots of quadratic equation $a{{x}^{2}}+bx+c$ is given by formula $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ . Now if $\alpha $ and β are the roots of the expression then $x-\alpha $ and $x-\beta $ are the factors of the given expression. Hence we can factorize any given quadratic.