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Factor.\newline2w2+5w+32w^2 + 5w + 3

Full solution

Q. Factor.\newline2w2+5w+32w^2 + 5w + 3
  1. Identify coefficients: Identify the coefficients aa, bb, and cc in the quadratic expression 2w2+5w+32w^2 + 5w + 3 by comparing it to the standard form ax2+bx+cax^2 + bx + c.a=2a = 2, b=5b = 5, c=3c = 3
  2. Find numbers and sum: Find two numbers that multiply to aca*c (which is 23=62*3 = 6) and add up to bb (which is 55).\newlineThe numbers that satisfy these conditions are 22 and 33 because 23=62*3 = 6 and 2+3=52+3 = 5.
  3. Rewrite middle term: Rewrite the middle term of the quadratic expression using the two numbers found in the previous step. \newline2w2+5w+32w^2 + 5w + 3 can be rewritten as 2w2+2w+3w+32w^2 + 2w + 3w + 3.
  4. Factor by grouping: Factor by grouping. Group the first two terms together and the last two terms together.\newline(2w2+2w)+(3w+3)(2w^2 + 2w) + (3w + 3)
  5. Factor out common factor: Factor out the greatest common factor from each group.\newline2w(w+1)+3(w+1)2w(w + 1) + 3(w + 1)
  6. Final factorization: Since both groups contain the common factor (w+1)(w + 1), factor this out.(2w+3)(w+1)(2w + 3)(w + 1)