Question

Multiply the expression. see attached picture of problem

Answer 1

Consider each polynom separately:

1)
`3x^2+2x-21`

Since

`D=2^2-4\cdot 3\cdot (-21)=4+252=256 \\ √(D)=16  \\ x_{1,2= (-2\pm 16)/(6) } =-3; (7)/(3)`, then
`3x^2+2x-21=3(x+3)(x-(7)/(3))=(x+3)(3x-7)`.


2)

`-2x^2-2x+12=2(x+3)(x-2) \\ D=(-2)^2-4\cdot(-2)\cdot 12=4+96=100 \\ √(D)=10 \\ x_(1,2)=(2\pm 10)/(-4) =-3;2`.

Then
`(3x^2+2x-21)/(-2x^2-2x+12) = ((x+3)(3x-7))/(2(x+3)(x-2) ) = (3x-7)/(2(x-2))`.

Similarly,3)

`2x^2+25x+63=2(x+9)(x+(7)/(2) ) =(x+9)(2x+7)\\ D=25^2-4\cdot 2\cdot 63=625-504=121 \\ √(D)=11 \\ x_(1,2)= (-25\pm 11)/(4) =-9;- (7)/(2)`and

4)

`6x^2+7x-49=6(x- (7)/(3))(x+ (7)/(2))=(3x-7)(2x+7) \\ D=7^2-4\cdot6\cdot(-49)=1225 \\ √(D)= 35 \\ x_(1,2)=(-7\pm35)/(12)= (7)/(3) ; -(7)/(2)`

Then

`(2x^2+25x+63)/(6x^2+7x-49) = ((x+9)(2x+7))/((3x-7)(2x+7) ) = (x+9)/(3x-7)`.
Now multiplication becomes easier:
`(3x^2+2x-21)/(-2x^2-2x+12)\cdot (2x^2+25x+63)/(6x^2+7x-49) = (3x-7)/(2(x-2))\cdot (x+9)/(3x-7) = (x+9)/(2(x-2))`

You obtain answer
`(x+9)/(2x-4)`, where a=1, b=9, c=2, d=-4.