Factors of the given polynomial

Splitting the middle term is a method for factoring a quadratic of the form ax2 + bx + c: find two numbers p and q such that their product p × q equals a × c and their sum p + q equals b. Rewriting the middle term bx as px + qx turns the trinomial into four terms that share a common binomial factor, which is then pulled out by grouping.

Applying this to 4√3x2 + 5x − 2√3:

1. Identify the coefficients: a = 4√3, b = 5, c = −2√3.

2. Compute a × c = 4√3 × (−2√3) = −8 × 3 = −24, and look for two numbers whose product is −24 and whose sum is 5.

3. The numbers 8 and −3 satisfy this: 8 × (−3) = −24 and 8 + (−3) = 5.

4. Split the middle term: 4√3x2 + 5x − 2√3 = 4√3x2 + 8x − 3x − 2√3.

5. Group the four terms: (4√3x2 + 8x) − (3x + 2√3).

6. Factor each group: 4x(√3x + 2) − √3(√3x + 2) — since 3x = √3 × √3x, pulling √3 out of 3x leaves √3x.

7. Factor out the common binomial (√3x + 2): this gives (4x − √3)(√3x + 2).

Cross-check by expanding (√3x + 2)(4x − √3) back out: 4x·√3x + 4x·2 − √3·√3x − √3·2 = 4√3x2 + 8x − 3x − 2√3 = 4√3x2 + 5x − 2√3, which matches the original polynomial term for term — confirming the factorization (√3x + 2)(4x − √3).