To factor the polynomial [tex]\(16x^2 + 40x + 25\)[/tex] completely, we need to find factors that multiply to give the polynomial. Let's proceed step-by-step:
1. Identify the given polynomial:
The polynomial given is [tex]\(16x^2 + 40x + 25\)[/tex].
2. Identify the coefficients:
- The coefficient of [tex]\(x^2\)[/tex] is [tex]\(16\)[/tex].
- The coefficient of [tex]\(x\)[/tex] is [tex]\(40\)[/tex].
- The constant term is [tex]\(25\)[/tex].
3. Find the factors:
To factor this polynomial, we need to look for two binomials that multiply to give back the original quadratic polynomial:
[tex]\[
(ax + b)(cx + d)
\][/tex]
4. Trial and error (or inspection) to find suitable factors (or using a more systematic approach):
We look for numbers that multiply to give the product of the leading coefficient (16) and the constant term (25). The product is [tex]\(16 \times 25 = 400\)[/tex].
5. Identify pairs:
Let's find a pair of factors [tex]\( (ax + b) \)[/tex] and [tex]\( (cx + d) \)[/tex] whose cross-multiplication gives the middle term 40x. Notice that the pair:
[tex]\[
(4x + 5)\text{ and }(4x + 5)
\][/tex]
works because:
[tex]\[
(4x + 5)(4x + 5) = 16x^2 + 20x + 20x + 25 = 16x^2 + 40x + 25
\][/tex]
6. Verify the factors:
[tex]\((4x + 5)^2 = (4x + 5)(4x + 5)\)[/tex]
Expanding this:
[tex]\[
(4x + 5)(4x + 5) = 16x^2 + 20x + 20x + 25 = 16x^2 + 40x + 25
\][/tex]
Thus, the polynomial [tex]\(16x^2 + 40x + 25\)[/tex] factors completely as:
[tex]\[
(4x + 5)^2 \text{ or } (4x + 5)(4x + 5).
\][/tex]
Therefore, the correct factored form from the given choices is:
[tex]\[
(4x + 5)(4x + 5)
\][/tex]
