Sure, let's factor the polynomial [tex]\(4x^2 + 25x + 6\)[/tex] completely.
1. Identify the polynomial to be factored:
[tex]\[
4x^2 + 25x + 6
\][/tex]
2. Look for a pair of binomials whose product gives the polynomial. We need two binomials of the form:
[tex]\[
(ax + b)(cx + d) = 4x^2 + 25x + 6
\][/tex]
Here, the product of the leading coefficients of [tex]\(ax\)[/tex] and [tex]\(cx\)[/tex] should equal the leading coefficient of the original polynomial ([tex]\(4x^2\)[/tex]), and the product of the constants [tex]\(b\)[/tex] and [tex]\(d\)[/tex] should equal the constant term ([tex]\(6\)[/tex]).
3. Set up the binomials considering possible factor pairs of the leading coefficient and the constant term:
[tex]\[
(4x + 1)(x + 6)
\][/tex]
4. Verify the factorization by expansion to ensure it results in the original polynomial:
Expanding [tex]\((4x + 1)(x + 6)\)[/tex]:
[tex]\[
(4x + 1)(x + 6) = 4x \cdot x + 4x \cdot 6 + 1 \cdot x + 1 \cdot 6
\][/tex]
[tex]\[
= 4x^2 + 24x + x + 6
\][/tex]
[tex]\[
= 4x^2 + 25x + 6
\][/tex]
The expanded form matches the original polynomial.
Therefore, the complete factorization of [tex]\(4x^2 + 25x + 6\)[/tex] is:
[tex]\[
(4x + 1)(x + 6)
\][/tex]
