Theory:

The square root of a polynomial using factorization method can be used only if the degrees of the polynomial are smaller.
The procedure to find the square root of a polynomial using factorization method is given by:
 
Step 1: If the given polynomial is a quadratic equation, then factorize them by splitting the middle terms or factorize them using algebraic identities.
 
Step 2: Take square root of the factorized terms.
Let us understand how to find the square root of a polynomial by the factorization method by the following examples.
Example:
1. Find the square root of \(144(x - a)^2(x -b)^{10}\).
 
Solution:
 
\(\sqrt{144(x - a)^2(x -b)^{10}} = 12|(x - a)(x -b)^5|\)
 
2. Find the square root of \((5x - 2y)^2 + 40xy\).
 
Solution:
 
\(\sqrt{(5x - 2y)^2 + 40xy} = \sqrt{25x^2 - 20xy + 4y^2 + 40xy}\)
 
\(= \sqrt{25x^2 + 20xy + 4y^2}\)
 
\(= \sqrt{(5x)^2 + 2(5x)(2y) + (2y)^2}\)
 
\(= \sqrt{(5x + 2y)^2}\)
 
\(= |(5x + 2y)|\)
 
3. Find the square root of \((6x^2 + 7x - 5)(2x^2 + 9x - 5)(3x^2 + 20x + 25)\)
 
Solution:
 
Let us factorize the polynomials.
 
\(6x^2 + 7x - 5 = 6x^2 + 10x - 3x - 5\)
 
\(= 2x(3x + 5) - 1(3x + 5)\)
 
\(= (3x + 5)(2x - 1)\)
 
\(2x^2 + 9x - 5 = 2x^2 + 10x - x - 5\)
 
\(= 2x(x + 5) - 1(x + 5)\)
 
\(= (2x - 1)(x + 5)\)
 
\(3x^2 + 20x + 25 = 3x^2 + 15x + 5x + 25\)
 
\(= 3x(x + 5) + 5(x + 5)\)
 
\(= (x + 5)(3x + 5)\)
 
\(\sqrt{(6x^2 + 7x - 5)(2x^2 + 9x - 5)(3x^2 + 20x + 25)} = \sqrt{(3x + 5)(2x - 1)(2x - 1)(x + 5)(x + 5)(3x + 5)}\)
 
\(= \sqrt{(3x + 5)^2(2x - 1)^2(x + 5)^2}\)
 
\(= (3x + 5)(2x - 1)(x + 5)\)