Determine what descartes rule of signs says about the number of positive and negative real roots for the polynomial function. p(x)=x^2+5x+6

descartes rule of signs says the max no. of +ve real roots

= no. of changes in sign of the coefficients of a polynomial

so max no. of +ve real roots for

P(x)=x^2+5x+6 which has no change in sign of coefficients (1,5,6)

is zero

P(-x)=(-x)^2+5(-x)+6

=x^2-5x+6 which has 2 changes in sign (1,-5,6)

max no. of -ve real roots is two

There are no changes in sign for p(x) = x^2 + 5x + 6 so no positive roots.

Replace x by (-x):-

P(-x)  = + (-x)^2 - 5x + 6

Here we have 2 changes of sign so the rule says there might be 2 negative roots

Normal way:
Use the distributive property
-4x - 2 = 26
-4x = 28
x = -7

And the lousy way:
Try making x every single number from -100 to 100. Stop when you find the number that makes the equation true.

Trying -100: -2(2(-100)+1)=398
Trying -99: -2(2(-99)+1)=394
Trying -98: -2(2(-98)+1)=390
Trying -97: -2(2(-97)+1)=386
Trying -96: -2(2(-96)+1)=382
Trying -95: -2(2(-95)+1)=378
Trying -94: -2(2(-94)+1)=374
Trying -93: -2(2(-93)+1)=370
Trying -92: -2(2(-92)+1)=366
Trying -91: -2(2(-91)+1)=362
Trying -90: -2(2(-90)+1)=358
Trying -89: -2(2(-89)+1)=354
Trying -88: -2(2(-88)+1)=350
Trying -87: -2(2(-87)+1)=346
Trying -86: -2(2(-86)+1)=342
Trying -85: -2(2(-85)+1)=338
Trying -84: -2(2(-84)+1)=334
Trying -83: -2(2(-83)+1)=330
Trying -82: -2(2(-82)+1)=326
Trying -81: -2(2(-81)+1)=322
Trying -80: -2(2(-80)+1)=318
Trying -79: -2(2(-79)+1)=314
Trying -78: -2(2(-78)+1)=310
Trying -77: -2(2(-77)+1)=306
Trying -76: -2(2(-76)+1)=302
Trying -75: -2(2(-75)+1)=298
Trying -74: -2(2(-74)+1)=294
Trying -73: -2(2(-73)+1)=290
Trying -72: -2(2(-72)+1)=286
Trying -71: -2(2(-71)+1)=282
Trying -70: -2(2(-70)+1)=278
Trying -69: -2(2(-69)+1)=274
Trying -68: -2(2(-68)+1)=270
Trying -67: -2(2(-67)+1)=266
Trying -66: -2(2(-66)+1)=262
Trying -65: -2(2(-65)+1)=258
Trying -64: -2(2(-64)+1)=254
Trying -63: -2(2(-63)+1)=250
Trying -62: -2(2(-62)+1)=246
Trying -61: -2(2(-61)+1)=242
Trying -60: -2(2(-60)+1)=238
Trying -59: -2(2(-59)+1)=234
Trying -58: -2(2(-58)+1)=230
Trying -57: -2(2(-57)+1)=226
Trying -56: -2(2(-56)+1)=222
Trying -55: -2(2(-55)+1)=218
Trying -54: -2(2(-54)+1)=214
Trying -53: -2(2(-53)+1)=210
Trying -52: -2(2(-52)+1)=206
Trying -51: -2(2(-51)+1)=202
Trying -50: -2(2(-50)+1)=198
Trying -49: -2(2(-49)+1)=194
Trying -48: -2(2(-48)+1)=190
Trying -47: -2(2(-47)+1)=186
Trying -46: -2(2(-46)+1)=182
Trying -45: -2(2(-45)+1)=178
Trying -44: -2(2(-44)+1)=174
Trying -43: -2(2(-43)+1)=170
Trying -42: -2(2(-42)+1)=166
Trying -41: -2(2(-41)+1)=162
Trying -40: -2(2(-40)+1)=158
Trying -39: -2(2(-39)+1)=154
Trying -38: -2(2(-38)+1)=150
Trying -37: -2(2(-37)+1)=146
Trying -36: -2(2(-36)+1)=142
Trying -35: -2(2(-35)+1)=138
Trying -34: -2(2(-34)+1)=134
Trying -33: -2(2(-33)+1)=130
Trying -32: -2(2(-32)+1)=126
Trying -31: -2(2(-31)+1)=122
Trying -30: -2(2(-30)+1)=118
Trying -29: -2(2(-29)+1)=114
Trying -28: -2(2(-28)+1)=110
Trying -27: -2(2(-27)+1)=106
Trying -26: -2(2(-26)+1)=102
Trying -25: -2(2(-25)+1)=98
Trying -24: -2(2(-24)+1)=94
Trying -23: -2(2(-23)+1)=90
Trying -22: -2(2(-22)+1)=86
Trying -21: -2(2(-21)+1)=82
Trying -20: -2(2(-20)+1)=78
Trying -19: -2(2(-19)+1)=74
Trying -18: -2(2(-18)+1)=70
Trying -17: -2(2(-17)+1)=66
Trying -16: -2(2(-16)+1)=62
Trying -15: -2(2(-15)+1)=58
Trying -14: -2(2(-14)+1)=54
Trying -13: -2(2(-13)+1)=50
Trying -12: -2(2(-12)+1)=46
Trying -11: -2(2(-11)+1)=42
Trying -10: -2(2(-10)+1)=38
Trying -9: -2(2(-9)+1)=34
Trying -8: -2(2(-8)+1)=30
Trying -7: -2(2(-7)+1)=26
We got it! the answer is -7