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.

$$6x^2 + 7x - 5 = 6x^2 + 10x - 3x - 5$$

$$= 2x(3x + 5) - 1(3x + 5)$$

$$= (3x + 5)(2x - 1)$$

$$2x^2 + 9x - 5 = 2x^2 + 10x - x - 5$$

$$= 2x(x + 5) - 1(x + 5)$$

$$= (2x - 1)(x + 5)$$

$$3x^2 + 20x + 25 = 3x^2 + 15x + 5x + 25$$

$$= 3x(x + 5) + 5(x + 5)$$

$$= (x + 5)(3x + 5)$$

$$\sqrt{(6x^2 + 7x - 5)(2x^2 + 9x - 5)(3x^2 + 20x + 25)} = \sqrt{(3x + 5)(2x - 1)(2x - 1)(x + 5)(x + 5)(3x + 5)}$$

$$= \sqrt{(3x + 5)^2(2x - 1)^2(x + 5)^2}$$

$$= (3x + 5)(2x - 1)(x + 5)$$