[tex]x^3+5x^2+9x+45=x^2(x+5)+9(x+5)=(x+5)(x^2+9)=(*)\\\\x^2+9=x^2+3^2=x^2-(-1)(3^2)=x^2-(i^2)(3^2)=x^2-(3i)^2\\\\i=\sqrt{-1}\to i^2=-1\\\\(*)=(x+5)[x^2-(3i)^2]=(x+5)(x-3i)(x+3i)\\\\Used:\ a^2-b^2=(a-b)(a+b)\\\\Answer:\ x^3+5x^2+9x+45=(x+5)(x-3i)(x+3i)[/tex]
The expression after completely factorizing is given by:
[tex]x^3+5x^2+9x+45=(x+5)(x+3i)(x-3i)[/tex]
We are given an algebraic expression in terms of the variable x as follows:
[tex]x^3+5x^2+9x+45[/tex]
We are asked to factor the expression completely.
i.e. the expression is written as follows:
[tex]x^3+5x^2+9x+45=x^2(x+5)+9(x+5)[/tex]
which is nothing but:
[tex]x^3+5x^2+9x+45=(x^2+9)(x+5)[/tex]
Now, we know that any quadratic equation of the type:
[tex]ax^2+bx+c=0[/tex]
has solution by the quadratic formula as:
[tex]x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
Here we have to find the solution of the quadratic equation:
[tex]x^2+9=0[/tex]
i.e.
[tex]a=1,\ b=0\ and\ c=9[/tex]
Hence, the solution is given by:
[tex]x=\dfrac{-0\pm \sqrt{0^2-4\times 1\times 9}}{2\times 1}\\\\x=\dfrac{\pm \sqrt{-36}}{2}\\\\x=\dfrac{\pm 6i}{2}\\\\x=\pm 3i\\\\x=3i\ and\ x=-3i\\\\x-3i=0\ and\ x+3i=0[/tex]
Hence, we have the expression as follows:
[tex]x^3+5x^2+9x+45=(x+5)(x+3i)(x-3i)[/tex]
