1) $(x+3)(x+5) \quad x^2+8 x+15$
2) $(2 x+1)(2 x+5) 4 x^2+12 x+5$
3) $(3 x-2)(x+4) \quad 3 x^2+10 x-8$
4) $(5 x+2)(x-3) \quad x x^2-13 x-6$
5) $(x-9)(2 x-3) \quad 2 x^2-21 x+27

The expansions of the first three expressions and the last expression are correct, matching the given forms exactly. However, the fourth expression has a discrepancy, as the provided form does not match the correct expansion. The accurate expanded form for the fourth expression should be 5 x 2 − 13 x − 6 .
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Answered by Anonymous | 2025-07-04

Expands each of the 5 given expressions.
Compares the expanded forms with the given forms.
Identifies that the fourth expression, ( 5 x + 2 ) ( x − 3 ) , is incorrectly expanded in the problem.
Concludes that the expansion of ( 5 x + 2 ) ( x − 3 ) should be 5 x 2 − 13 x − 6 , but the problem states x x 2 − 13 x − 6 .

Explanation

Problem Analysis We are given 5 expressions of the form ( a x + b ) ( c x + d ) and their expanded forms. Our objective is to verify if the expanded forms are correct. We will expand each expression and compare it with the given expanded form.

Expanding the First Expression



( x + 3 ) ( x + 5 ) = x 2 + 5 x + 3 x + 15 = x 2 + 8 x + 15 . The given expanded form is x 2 + 8 x + 15 , which matches our calculation.


Expanding the Second Expression


( 2 x + 1 ) ( 2 x + 5 ) = 4 x 2 + 10 x + 2 x + 5 = 4 x 2 + 12 x + 5 . The given expanded form is 4 x 2 + 12 x + 5 , which matches our calculation.


Expanding the Third Expression


( 3 x − 2 ) ( x + 4 ) = 3 x 2 + 12 x − 2 x − 8 = 3 x 2 + 10 x − 8 . The given expanded form is 3 x 2 + 10 x − 8 , which matches our calculation.


Expanding the Fourth Expression


( 5 x + 2 ) ( x − 3 ) = 5 x 2 − 15 x + 2 x − 6 = 5 x 2 − 13 x − 6 . The given expanded form is x x 2 − 13 x − 6 . This does not match our calculation. The correct expanded form should be 5 x 2 − 13 x − 6 .


Expanding the Fifth Expression


( x − 9 ) ( 2 x − 3 ) = 2 x 2 − 3 x − 18 x + 27 = 2 x 2 − 21 x + 27 . The given expanded form is 2 x 2 − 21 x + 27 , which matches our calculation.


Conclusion Comparing the calculated expanded forms with the given expanded forms, we find that the fourth expression has a discrepancy. The given expanded form for ( 5 x + 2 ) ( x − 3 ) is incorrect.

Examples
Understanding how to expand expressions like these is fundamental in algebra and has many practical applications. For example, if you're designing a rectangular garden where the length is ( x + 5 ) meters and the width is ( x + 3 ) meters, expanding ( x + 3 ) ( x + 5 ) gives you the area of the garden, x 2 + 8 x + 15 square meters. This skill is also crucial in physics for calculating areas and volumes, and in economics for modeling revenue and cost functions.

Answered by GinnyAnswer | 2025-07-04