Answer:
Explanation:
a. Given
and
![6x^2-2x+5](https://img.qammunity.org/2021/formulas/mathematics/middle-school/s9q4k0eqrwacpvnc0wwp9ohzphemhxio3v.png)
Required
Product
The solution is as follows
First, put both expression in different brackets
![(3x-9)(6x^2-2x+5)](https://img.qammunity.org/2021/formulas/mathematics/middle-school/b2oim8bsaj1y3tfle2bb9yl4hfznnovsno.png)
Expand
bracket by
![(3x-9)](https://img.qammunity.org/2021/formulas/mathematics/middle-school/4racvcrha6llxyz9jwcjwf2mi4fo8vh2vx.png)
To do that. we first multiply
by 3x then by -9. This is done as follows
![3x (6x^2-2x+5) - 9(6x^2-2x+5)](https://img.qammunity.org/2021/formulas/mathematics/middle-school/d5e7og0znmg6yx5fpmx52uwqo9wq6ij2gf.png)
Now, we proceed to opening the bracket
![(3x * 6x^2-3x * 2x+3x *5) - (9* 6x^2-9* 2x+9* 5)](https://img.qammunity.org/2021/formulas/mathematics/middle-school/foaz95mcxxo09q201r4s7faykxwyg4yv4u.png)
![(18x^3 -6x^2+ 15x) - (54x^2 - 18x +45)](https://img.qammunity.org/2021/formulas/mathematics/middle-school/2p3toli5p2pmy44hvv8lzbuqolbf3n5okz.png)
Open both brackets
![18x^3 -6x^2+ 15x -54x^2 + 18x -45](https://img.qammunity.org/2021/formulas/mathematics/middle-school/ovzim6b4ag14n0o72v1zoc8ltz6w0ge1ub.png)
Collect like terms
![18x^3 -6x^2 -54x^2 + 15x + 18x -45](https://img.qammunity.org/2021/formulas/mathematics/middle-school/dq0e8g56v8aatgj3b2nuoe1aqufy3q2bec.png)
![18x^3 -60x^2 + 33x - 45](https://img.qammunity.org/2021/formulas/mathematics/middle-school/rbw55ktd0ts5ob899t2h8quwvumt34g6s8.png)
Hence, the product of
and
is
![18x^3 -60x^2 + 33x - 45](https://img.qammunity.org/2021/formulas/mathematics/middle-school/rbw55ktd0ts5ob899t2h8quwvumt34g6s8.png)
b. Given
and
![6x^2-2x+5](https://img.qammunity.org/2021/formulas/mathematics/middle-school/s9q4k0eqrwacpvnc0wwp9ohzphemhxio3v.png)
and
![6x^2-2x+5](https://img.qammunity.org/2021/formulas/mathematics/middle-school/s9q4k0eqrwacpvnc0wwp9ohzphemhxio3v.png)
Required
Are their products equal?
To check if they are equal or not, we find the product of both and compare the solutions
We've already solved for
and
in the (a) part of this exercise, so we move to the product of
and
![6x^2-2x+5](https://img.qammunity.org/2021/formulas/mathematics/middle-school/s9q4k0eqrwacpvnc0wwp9ohzphemhxio3v.png)
The solution is as follows
First, put both expression in different brackets
![(9x-3)(6x^2-2x+5)](https://img.qammunity.org/2021/formulas/mathematics/middle-school/lk4dr2sx0fggj3qp8dypkf7faakjjxyw23.png)
Expand
bracket by
![(9x-3)](https://img.qammunity.org/2021/formulas/mathematics/middle-school/3zubdqvcy7u665hzyxxle3ps2inpci156h.png)
To do that. we first multiply
by 9x then by -3. This is done as follows
![9x (6x^2-2x+5) - 3(6x^2-2x+5)](https://img.qammunity.org/2021/formulas/mathematics/middle-school/w90kicbksp165tiiywufldbfspha57ue85.png)
Now, we proceed to opening the bracket
![(9x * 6x^2 - 9x * 2x + 9x *5) - (3* 6x^2 - 3 * 2x + 3* 5)](https://img.qammunity.org/2021/formulas/mathematics/middle-school/b7wuq3zfc6qq3c9dnp8y88ecw1uudgfoku.png)
![(54x^3 - 18x^2+ 45x) - (18x^2 - 6x +15)](https://img.qammunity.org/2021/formulas/mathematics/middle-school/r5wra6vypfljt4s82kkxv0jwja92p2888l.png)
Open both brackets
![54x^3 - 18x^2+ 45x -18x^2 + 6x -15](https://img.qammunity.org/2021/formulas/mathematics/middle-school/syxaxhans1du1x1796neyz8ks4r6y5hbnv.png)
Collect like terms
![54x^3 - 18x^2 -18x^2 + 45x + 6x -15](https://img.qammunity.org/2021/formulas/mathematics/middle-school/6spdmc5a6ai5fh9ed4l2lx34vu1sa1rvfv.png)
![54x^3 - 36x^2 + 51x -15](https://img.qammunity.org/2021/formulas/mathematics/middle-school/g1minczywdtos9o1wmbrq4xwxmm1s957cl.png)
Now, we compare both answers
Is
![18x^3 -60x^2 + 33x - 45](https://img.qammunity.org/2021/formulas/mathematics/middle-school/rbw55ktd0ts5ob899t2h8quwvumt34g6s8.png)
equal to
![54x^3 - 36x^2 + 51x -15](https://img.qammunity.org/2021/formulas/mathematics/middle-school/g1minczywdtos9o1wmbrq4xwxmm1s957cl.png)
No, they're not.
Reason is that, for both expressions to be equal, we must have the same expression after expanding both of them