Q. What is the factored form of \(8x^{2}+12x\)?

Problem: Factor: \(8x^{2}+12x\)

Step 1 — Identify the greatest common factor (GCF) of the coefficients and variables.

• Coefficients: \(\gcd(8,12)=4\).
• Variables: the terms are \(8x^{2}\) and \(12x\). The smallest power of \(x\) present is \(x^{1}\), so the variable part of the GCF is \(x\).
• Therefore the overall GCF is \(4x\).

Step 2 — Divide each term by the GCF and factor it out.

• \(\dfrac{8x^{2}}{4x}=2x\).
• \(\dfrac{12x}{4x}=3\).
• Factor out \(4x\) from the original expression:
  \[
  8x^{2}+12x=4x\bigl(2x+3\bigr).
  \]

Step 3 — Verify by distributing to confirm equality.

• Distribute \(4x\) over \(\bigl(2x+3\bigr)\):
  \[
  4x\cdot 2x=8x^{2},\qquad 4x\cdot 3=12x.
  \]
• Sum of distributed terms: \(8x^{2}+12x\).
• Thus the factorization is correct.

Final factored form:

\[
8x^{2}+12x=4x\bigl(2x+3\bigr)
\]