To factor the quadratic expression [tex]\(4x^2 + 15x + 14\)[/tex] fully, we need to find two binomials whose product gives the original quadratic expression.
The expression provided is a quadratic equation in the general form [tex]\(ax^2 + bx + c\)[/tex], where [tex]\(a = 4\)[/tex], [tex]\(b = 15\)[/tex], and [tex]\(c = 14\)[/tex]. The goal is to express the quadratic as a product of two binomials:
[tex]\[
(ax + b_1)(cx + d_1)
\][/tex]
Here is a step-by-step solution to factor [tex]\(4x^2 + 15x + 14\)[/tex]:
1. Identify the terms:
- The coefficient of [tex]\(x^2\)[/tex] is 4.
- The coefficient of [tex]\(x\)[/tex] is 15.
- The constant term is 14.
2. Set up the binomials:
To find [tex]\((ax + b_1)(cx + d_1)\)[/tex] such that their product is [tex]\(4x^2 + 15x + 14\)[/tex].
3. Find factors of the constant term (14):
The factors of 14 are [tex]\((1, 14)\)[/tex], [tex]\((2, 7)\)[/tex], [tex]\((7, 2)\)[/tex], and [tex]\((14, 1)\)[/tex].
4. Consider the middle term split:
We need to split the middle term 15x into two parts whose coefficients multiply to give [tex]\(4 \cdot 14 = 56\)[/tex].
5. Search for appropriate pairs:
We need numbers whose product is 56 and whose sum is 15.
The correct pair is [tex]\(7\)[/tex] and [tex]\(8\)[/tex] since:
[tex]\[
7 + 8 = 15
\][/tex]
[tex]\[
7 \cdot 8 = 56
\][/tex]
6. Construct the binomials:
We can rewrite 15x as follows:
[tex]\[
4x^2 + 8x + 7x + 14
\][/tex]
7. Factor by grouping:
Group the terms as:
[tex]\[
(4x^2 + 8x) + (7x + 14)
\][/tex]
8. Factor out the common factors in each group:
[tex]\[
4x(x + 2) + 7(x + 2)
\][/tex]
9. Factor out the common binomial factor (x + 2):
[tex]\[
(4x + 7)(x + 2)
\][/tex]
Therefore, the fully factored form of [tex]\(4x^2 + 15x + 14\)[/tex] is:
[tex]\[
(4x + 7)(x + 2)
\][/tex]
Hence, the correct option is:
(4x + 7)(x + 2).
