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Find factors of the quadratic expression: 3x2+15x+123x^{2}+15x+12\newline(?x+?)(x+?)(?x+?)(x+?)

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Full solution

Q. Find factors of the quadratic expression: 3x2+15x+123x^{2}+15x+12\newline(?x+?)(x+?)(?x+?)(x+?)
  1. Identify Numbers: We need to factor the quadratic expression 3x2+15x+123x^2 + 15x + 12 into the form (ax+b)(cx+d)(ax + b)(cx + d), where aa, bb, cc, and dd are numbers we need to find.
  2. Find Multiplication Result: First, we look for two numbers that multiply to give the product of the coefficient of x2x^2 (which is 33) and the constant term (which is 1212). So we need two numbers that multiply to 3×12=363 \times 12 = 36.
  3. Determine Sum: Next, we also need these two numbers to add up to the coefficient of the xx term, which is 1515.
  4. Use Identified Numbers: The two numbers that multiply to 3636 and add up to 1515 are 33 and 1212.
  5. Write Quadratic Expression: Now we can write the quadratic expression using these two numbers: 3x2+3x+12x+123x^2 + 3x + 12x + 12.
  6. Group Terms: We can factor by grouping. First, we group the terms: 3x2+3x3x^2 + 3x + 12x+1212x + 12.
  7. Factor Out Common Factor: Factor out the greatest common factor from each group: 3x(x+1)+12(x+1)3x(x + 1) + 12(x + 1).
  8. Factor Out (x+1)(x + 1): Since both groups contain the factor (x+1)(x + 1), we can factor this out: (3x+12)(x+1)(3x + 12)(x + 1).
  9. Simplify First Term: Finally, we can simplify the first term by factoring out the common factor of 33: 3(x+4)(x+1)3(x + 4)(x + 1).

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