how do you factorise x squared +2x+3

1(2xsquared+3)

x^2 + 2x + 3

does not factor over the rational numbers

(there are no two rational numbers which when added yield 2 and which when multiplied yield 3)

However, if you made a typo and the real polynomial is x^2 + 2x -3, then you get (x+3)(x-1)

To factorize the quadratic expression x^2 + 2x + 3, we need to find two binomials that, when multiplied together, would give us the original quadratic expression.

One way to factorize it is by using the quadratic formula. The quadratic formula states that for any quadratic equation of the form ax^2 + bx + c = 0, the roots can be found using the formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, we have x^2 + 2x + 3 = 0, so a = 1, b = 2, and c = 3.

Let's substitute these values into the quadratic formula:

x = (-(2) ± √((2)^2 - 4(1)(3))) / (2(1))

Simplifying further:

x = (-2 ± √(4 - 12)) / 2
x = (-2 ± √(-8)) / 2

Since we have a negative value inside the square root, it means that the expression x^2 + 2x + 3 has no real roots. Therefore, it cannot be factored into linear terms with real coefficients.

However, we can still express it as the product of two binomials with complex coefficients. Using the quadratic formula, we can rewrite it as:

x = (-2 ± 2i√(2)) / 2

Simplifying further:

x = -1 ± i√(2)

Therefore, the factored form of x^2 + 2x + 3 is:

(x + (1 - i√(2)))(x + (1 + i√(2)))

Please note that factoring quadratics can sometimes be more complex, but this approach should work for most cases.