Solve for x in each of the following, x E C. (x+1)(x+5)(x+3) = -3the answer is x = -2, -7+- sqare root of 13 divided by 2

There is another method of solving the
problem.

We notice that the expression is a product of 3
factors:

(x+1)(x+5)(x+3)

Let's
consider -3 as a result of a product of 3 factors, too.

-3
= -1*1*3

Now, let's re-write the
expression:

(x+1)(x+5)(x+3) =
1*(-1)*3

We'll put the factor (x+1) =
1

x+1 = 1

We'll subtract 1
both sides:

x = 0

x+5 =
-1

We'll subtract 5 both sides, in order to isolate
x:

x = -6

x+3 =
3

x = 0

If, we'll substitute
the values into the expression (x+1)(x+5)(x+3) = -3, they verify
it.

Another way of solving, would be to remove the
brackets:

We'll calculate the product of the first and the
second factor:

(x+1)(x+5) = x^2 + 5x + x + 5 = x^2 + 6x +
5

Now, we'll multiply the result by
(x+3):

(x^2 + 6x + 5)(x+3) = x^3 + 3x^2 + 6x^2 + 18x + 5x +
15

We'll combine like
terms:

x^3 + 9x^2 + 23x + 15 + 3 =
0

 x^3 + 9x^2 + 23x + 18 =
0

Since -2 is a solution, we'll divide  x^3 + 9x^2 + 23x +
18 by (x+2) and the reminedr will be 0.

x^3 + 9x^2 + 23x +
18 = (x+2)(ax^2 + bx + c)

We'll remove the brackets from
the right side:

x^3 + 9x^2 + 23x + 18 = ax^3 + bx^2 + cx +
2ax^2 + 2bx + 2c

We'll combine like terms from the right
side and we'll factorize:

x^3 + 9x^2 + 23x + 18 = ax^3 +
x^2(b+2a) + x(c+2b) + 2c

Now, the polynomial from the left
side is identically with the polynomial from the right side, if and only if the
coefficients are equal:

a =
1

b+2a = 9 => b + 2 = 9 => b =
7

c + 2b = 23 => c + 14 = 23 => c =
9

The polynomial ax^2 + bx + c = x^2 + 7x +
9

Now, we'll determine the roots of the
quadratic:

 x^2 + 7x + 9 =
0

x1 =
[-7+sqrt(49-36)]/2

x1 =
(-7+sqrt13)/2

x2
=  (-7-sqrt13)/2

So, the roots
of (x+1)(x+5)(x+3) = -3 are {-2 ; (-7+sqrt13)/2 ;
(-7-sqrt13)/2}.