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Factor completely: (4x+1)(2x+5)-(4x+1)^(2)(4x+3) Answer:

Factor completely:\newline(4x+1)(2x+5)−(4x+1)2(4x+3) (4 x+1)(2 x+5)-(4 x+1)^{2}(4 x+3) \newlineAnswer:

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Full solution

Q. Factor completely:\newline(4x+1)(2x+5)−(4x+1)2(4x+3) (4 x+1)(2 x+5)-(4 x+1)^{2}(4 x+3) \newlineAnswer:
  1. Expand and Multiply: Distribute the square in the second term.\newlineWe need to expand (4x+1)2(4x+1)^2 to simplify the expression.\newline(4x+1)2=(4x+1)(4x+1)(4x+1)^2 = (4x+1)(4x+1)
  2. Multiply Expanded Square: Multiply the terms in the expanded square.\newlineNow we multiply the terms in the binomial (4x+1)(4x+1)(4x+1)(4x+1).\newline(4x+1)(4x+1)=16x2+4x+4x+1(4x+1)(4x+1) = 16x^2 + 4x + 4x + 1\newline=16x2+8x+1= 16x^2 + 8x + 1
  3. Distribute and Multiply: Multiply the expanded square by (4x+3)(4x+3).\newlineWe need to distribute (4x+3)(4x+3) across the terms in 16x2+8x+116x^2 + 8x + 1.\newline(16x2+8x+1)(4x+3)=16x2(4x)+16x2(3)+8x(4x)+8x(3)+1(4x)+1(3)(16x^2 + 8x + 1)(4x+3) = 16x^2(4x) + 16x^2(3) + 8x(4x) + 8x(3) + 1(4x) + 1(3)\newline=64x3+48x2+32x2+24x+4x+3= 64x^3 + 48x^2 + 32x^2 + 24x + 4x + 3\newline=64x3+80x2+28x+3= 64x^3 + 80x^2 + 28x + 3
  4. Subtract from First Term: Subtract the result from Step 33 from the first term (4x+1)(2x+5)(4x+1)(2x+5).\newlineNow we subtract the polynomial we just found from the first term in the original expression.\newline(4x+1)(2x+5)−(64x3+80x2+28x+3)(4x+1)(2x+5) - (64x^3 + 80x^2 + 28x + 3)\newline= 8x2+20x+2x+5−64x3−80x2−28x−38x^2 + 20x + 2x + 5 - 64x^3 - 80x^2 - 28x - 3\newline= 8x2+22x+5−64x3−80x2−28x−38x^2 + 22x + 5 - 64x^3 - 80x^2 - 28x - 3
  5. Combine Like Terms: Combine like terms.\newlineWe combine the like terms to simplify the expression.\newline8x2+22x+5−64x3−80x2−28x−38x^2 + 22x + 5 - 64x^3 - 80x^2 - 28x - 3\newline= −64x3+(8x2−80x2)+(22x−28x)+(5−3)-64x^3 + (8x^2 - 80x^2) + (22x - 28x) + (5 - 3)\newline= −64x3−72x2−6x+2-64x^3 - 72x^2 - 6x + 2
  6. Factor Out Common Factor: Factor out the common factor 4x+14x+1. We notice that the original expression had a common factor of 4x+14x+1 in both terms, so we factor it out from the simplified expression. (-64\)x^33 - 7272x^22 - 66x + 22 does not have a common factor of 4x+14x+1, so we cannot factor it out. This means we have made a mistake in our previous steps.

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