The factorization of $ \left( {4{x^2} + 8x + 3} \right) $ is

Hint: Here, comparing the given equation with quadratic polynomial we can see that given polynomial is a quadratic polynomial with variable x, split middle term and using factorization method factorize the given polynomial.

Complete step-by-step answer:
 A polynomial in the form of $ a{x^2} + bx + c,a \ne 0 $ is known as a quadratic polynomial. Degree of a quadratic polynomial is always 2.
Given polynomial is
 $ \left( {4{x^2} + 8x + 3} \right) $
Comparing the given polynomial with standard quadratic equation
 $ a{x^2} + bx + c,a \ne 0 $ ,
here, a = 4, b = 8 and c = 3.
Splitting middle term, putting 8x = 6x + 2x
 $ 4{x^2} + 8x + 3 = 4{x^2} + 6x + 2x + 3 $
$= 2x (2x + 3) + 1(2x + 3) = (2x + 3)(2x + 1)$
Thus, $ \left( {4{x^2} + 8x + 3} \right) = \left( {2x + 3} \right)\left( {2x + 1} \right) $
So, the correct answer is “ $ \left( {2x + 3} \right)\left( {2x + 1} \right)$”.

Note: Factorization method is preferred for these types of questions as it does not require so much calculation. But be careful while splitting middle term, product of middle term should be exactly equal to constant term. Find the values of x in terms of given constant, here a is considered as constant. Alternatively, a given polynomial can be factorized by considering a quadratic polynomial as a quadratic equation and equating it with 0. Find the values of x using quadratic formula, and write it in factor form.