Need help in math. College Algebra?

It is "irreducible" (prime) over the real numbers.

Remember the factor theorem? For a number "r", (x-r) will be a factor of a polynomial p(x) if and only if p(r)=0.

In your polynomial: 4x²+9≥9>0 (since x²≥0) for all real numbers "x". No zeros means no linear factors (wth real numbers for coefficients).

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However, IF you allow square roots of negative numbers:

   4x² + 9 = 0

   ⇒   4x² = -9

   ⇒   x² = -9/4

   ⇒   x = √(-9/4) or x = -√(-9/4)

Then [x-√(-9/4)] and [x+√(-9/4)] are factors, along with the constant "4".

Allowing square roots of negative numbers extends the real number system to form the "complex numbers", which you may not have studied yet. They are very important in a number of areas, including electronics. The "Fundamental Theorem of Algebra:" says that only linear polynomials are irreducible (prime) when you allow complex numbers.