`a)`

`P = -x^2 + 2xy - 4y^2 + 2x + 10y - 3`

`P = -(x^2 - 2xy + y^2 - 2x + 2y + 1) - 3y^2 + 12y - 4`

`P = -[(x-y)^2 - 2(x-y) + 1] - (3y^2 - 12y + 12) + 8`

`P = -(x - y - 1)^2 - 3(y^2 - 4y + 4) + 8`

`P = -(x - y - 1)^2 - 3(y - 2)^2 + 8`

Vì `-(x - y - 1)^2 <= 0 AA x, y`
Và `-3(y - 2)^2 <= 0 AA y`

Nên `P <= 8`

Dấu "`=`" xảy ra khi:

`{(x - y - 1 = 0),(y - 2 = 0):}`

`=> {(x - 2 - 1 = 0),(y = 2):}`

`=> {(x = 3),(y = 2):}`

Vậy `P_{"max"} = 8` khi `x = 3` và `y = 2`

`b)`

`Q = -x^2 + xy - y^2 - 2x + 4y + 11`

`Q = -(x^2 - xy + y^2 + 2x - 4y - 11)`
`Q = -[x^2 - x(y - 2)] - y^2 + 4y + 11`

`Q = -(x - (y-2)/2)^2 + (y^2 - 4y + 4)/4 - y^2 + 4y + 11`

`Q = -(x - y/2 + 1)^2 + 1/4 y^2 - y + 1 - y^2 + 4y + 11`

`Q = -(x - y/2 + 1)^2 - 3/4 y^2 + 3y + 12`

`Q = -(x - y/2 + 1)^2 - 3/4 (y^2 - 4y) + 12`

`Q = -(x - y/2 + 1)^2 - 3/4 (y^2 - 4y + 4 - 4) + 12`

`Q = -(x - y/2 + 1)^2 - 3/4 (y - 2)^2 + 3 + 12`

`Q = -(x - y/2 + 1)^2 - 3/4 (y - 2)^2 + 15`

Vì `-(x - y/2 + 1)^2 <= 0 AA x, y`
Và `-3/4 (y - 2)^2 <= 0 AA y`

Nên `Q <= 15`

Dấu "`=`" xảy ra khi:

`{(x - y/2 + 1 = 0),(y - 2 = 0):}`

`=> {(x - 2/2 + 1 = 0),(y = 2):}`

`=> {(x - 1 + 1 = 0),(y = 2):}`

`=> {(x = 0),(y = 2):}`

Vậy `Q_{"max"} = 15` khi `x = 0` và `y = 2`