Certainly! Let's find the completely factored form of the expression [tex]\(16x^2 + 8x + 32\)[/tex].
First, observe that each term in the expression has a common factor. The greatest common factor (GCD) of the coefficients 16, 8, and 32 is 8. Therefore, we can factor out 8:
[tex]\[16x^2 + 8x + 32 = 8(2x^2 + x + 4).\][/tex]
Now, we'll check if the expression inside the parenthesis, [tex]\(2x^2 + x + 4\)[/tex], can be factored further. To do this, we would look for two binomials that multiply to give [tex]\(2x^2 + x + 4\)[/tex], but upon inspection, [tex]\(2x^2 + x + 4\)[/tex] does not factorize further using simple methods.
Therefore, the completely factored form of the expression [tex]\(16x^2 + 8x + 32\)[/tex] is:
[tex]\[8(2x^2 + x + 4).\][/tex]
Comparing this with the given options:
- [tex]\(4(4x^2 + 2x + 8)\)[/tex] is not the same.
- [tex]\(4(12x^2 + 4x + 28)\)[/tex] is also not the same.
- [tex]\(8(2x^2 + x + 4)\)[/tex] matches our result.
- [tex]\(8x(8x^2 + x + 24)\)[/tex] is not the same.
Hence, the correct answer is:
[tex]\[8\left(2x^2 + x + 4\right).\][/tex]
