To factor the quadratic expression [tex]\(5x^2 + 23x + 12\)[/tex], we need to find two binomials whose product gives the original quadratic. Here’s a step-by-step approach:
1. Write down the quadratic expression:
[tex]\[
5x^2 + 23x + 12
\][/tex]
2. Understand the structure of the factors:
We are looking for two binomials of the form [tex]\((ax + b)(cx + d)\)[/tex] that expand to our quadratic expression. Specifically, we need to have:
[tex]\[
(ax + b)(cx + d) = 5x^2 + 23x + 12
\][/tex]
3. Identify the coefficients:
- The product of the constants (coefficients of [tex]\(x^0\)[/tex]) [tex]\(b \cdot d = 12\)[/tex]
- The product of the leading coefficients (coefficients of [tex]\(x^2\)[/tex]) [tex]\(a \cdot c = 5\)[/tex]
- The middle term (coefficient of [tex]\(x\)[/tex]) is formed by the sum of the products of the outer and inner terms in the binomials.
4. Guess and check for the middle term:
After finding appropriate pairs such that their outer and inner product sum up to the middle coefficient, we can determine the correct factors. Here, upon correctly factoring or calculating:
5. Write the factors:
The quadratic expression [tex]\(5x^2 + 23x + 12\)[/tex] can be factored as:
[tex]\[
(x + 4)(5x + 3)
\][/tex]
6. Verify the solution:
To ensure our factors are correct, we can expand [tex]\((x + 4)(5x + 3)\)[/tex]:
[tex]\[
(x + 4)(5x + 3) = x \cdot 5x + x \cdot 3 + 4 \cdot 5x + 4 \cdot 3
\][/tex]
Simplifying this:
[tex]\[
5x^2 + 3x + 20x + 12 = 5x^2 + 23x + 12
\][/tex]
This confirms our factorization is correct.
Thus, the factored form of the quadratic expression [tex]\(5x^2 + 23x + 12\)[/tex] is:
[tex]\[
(x + 4)(5x + 3)
\][/tex]
