How do you multiply and simplify #\frac { 6x ^ { 2} + 5x + 1} { 3x - 6} \cdot \frac { 2- x } { 4x ^ { 2} + 4x + 1}#?

1 Answer
Jim G.
Aug 2, 2017

#-(3x+1)/(3(2x+1)#

Explanation:

#"factorise the numerators/denominators of both fractions"#
#"and cancel any common factors"#

#•color(white)(x)6x^2+5x+1#

#"the factors which multiply to + 6 and add to +5"#
#"are + 3 and + 2"#

#"split the middle term and factorise by grouping"#

#rArr6x^2+3x+2x+1#

#=color(red)(3x)(2x+1)color(red)(+1)(2x+1)#

#=(2x+1)(color(red)(3x+1))#

#•color(white)(x)3x-6=3(x-2)#

#•color(white)(x)2-x=-(x-2)#

#•color(white)(x)4x^2+4x+1#

#"this is a perfect square and factorises as"#

#4x^2+4x+1=(2x+1)^2#

#rArr(6x^2+5x+1)/(3x-6)xx(2-x)/(4x^2+4x+1)#

#=((3x+1)cancel((color(red)(2x+1))))/(3cancel((color(blue)(x-2))))xx(-cancel((color(blue)(x-2))))/(cancel((color(red)(2x+1)))(2x+1))#

#=-(3x+1)/(3(2x+1))to(x!=-1/2)#

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