To factorise the given expression [tex]\( 3x^2 + 9x + 3 \)[/tex], we can follow these steps:
1. Identify the greatest common factor (GCF):
First, observe that all the terms in the expression share a common factor. In this case, you can see that 3 is a factor of each term in the expression. Therefore, factor out 3:
[tex]\[
3x^2 + 9x + 3 = 3(x^2 + 3x + 1)
\][/tex]
2. Factorise the quadratic expression inside the parentheses:
The expression inside the parentheses is [tex]\( x^2 + 3x + 1 \)[/tex]. We now need to check if this quadratic expression can be factored further. Quadratic expressions can often be factored into two binomials, but not always in simple forms where the coefficients are integers.
However, it turns out that [tex]\( x^2 + 3x + 1 \)[/tex] does not factor into simpler integer binomials, so the factorised form will still include this quadratic part.
3. Combine the factored terms:
Since the quadratic part [tex]\( x^2 + 3x + 1 \)[/tex] cannot be factored further without using complex numbers or irrational numbers, the factorised form of the original expression remains as:
[tex]\[
3(x^2 + 3x + 1)
\][/tex]
Thus, the factorisation of [tex]\( 3x^2 + 9x + 3 \)[/tex] is:
[tex]\[
3(x^2 + 3x + 1)
\][/tex]
