We'll have to solve the quadratic equation 7x^2+10x+3=0.

Since the request is not to apply the quadratic formula, we'll use factorization.

We notice that the coefficient of the middle term could be written as the sum of coefficients of the first and the last terms.

10 = 7 + 3

We'll multiply by x both sides:

10x = 7x + 3x

We'll re-write the equation:

7x^2 + 7x + 3x + 3 = 0

We'll combine the first 2 terms and the last 2 terms:

(7x^2 + 7x) + (3x + 3) = 0

We'll factorize by 7x the first pair of brackets and we'll factorize by 3 the second pair of brackets.

7x(x+1) + 3(x+1) = 0

We'll factorize by (x+1):

(x + 1)(7x + 3) = 0

We'll put each factor as zero:

x+1 = 0

**x1 = -1**

7x + 3 = 0

7x = -3

**x2 = -3/7**

**The roots of the quadratic equation are: {-1 ; -3/7}**

As we have to find the roots of the quadratic equation 7x^2 + 10x + 3 =0, without using the formula for finding the roots, we go about it this way.

First express 10x as two terms the product of which is 7x^2*3 = 21x^2.

We see that 10x can be written as 7x + 3x and 7x*3x = 21x^2

So we write the given equation as:

7x^2 + 10x + 3 =0

=> 7x^2 + 7x + 3x + 3 =0

=> 7x ( x+1 ) + 3(x+1) =0

=> (7x+3)(x+1) =0

=> 7x+3 = 0 or x+1 =0

=> x = -3/7 or x= -1

**Therefore the values of x we get are -1 and -3/7**.