Solution:

 \(2x^{2} + x - 1\)
= \(2x^{2} + 2x - x - 1\)
= \(2x\underline{(x + 1)} - 1\underline{(x + 1)}\)
= \((x + 1)(2x - 1)\)

Solution:

 \(2m^{2} + 5m - 3\)
= \(2m^{2} + 6m - m - 3\)
= \(2m\underline{(m + 3)} - 1\underline{(m + 3)}\)
= \((m + 3)(2m - 1)\)

Solution:

 \(12x^{2} + 61x + 77\)
= \(12x^{2} + 33x + 28x + 77\)
= \(3x\underline{(4x + 11)} + 7\underline{(4x + 11)}\)
= \((4x + 11)(3x + 7)\)

Solution:

 \(3y^{2} - 2y - 1\)
= \(3y^{2} - 3y + y - 1\)
= \(3y\underline{(y - 1)} + 1\underline{(y - 1)}\)
= \((y - 1)(3y + 1)\)

Solution:

 \(\sqrt{3}x^{2} + 4x + \sqrt{3}\)
= \(\sqrt{3}x^{2} + 3x + x + \sqrt{3}\)
= \(\sqrt{3}x\underline{(x + \sqrt{3})}) + 1\underline(\underline{(x + \sqrt{3})})\)
= \((x + \sqrt{3})(\sqrt{3}x + 1)\)

Solution:

 \(\displaystyle \frac{1}{2}x^{2} - 3x + 4\)

= \(\displaystyle \frac{1}{2}x^{2} - 2x - x + 4\)

= \(\displaystyle \frac{1}{2}x\underline{(x - 4)} - 1\underline{(x - 4)}\)

= \(\displaystyle (x - 4)\left(\frac{1}{2}x - 1\right)\)

Solution:

\(\bbox[yellow, 1pt, border: 2px solid red]{{(x^{2} - x)}}^{2} - 8\bbox[yellow, 1pt, border: 2px solid red]{{(x^{2} - x)}} + 12\)

Let, \((x^{2} - x) = m\)
∴ The given polynomial
= \(m^{2} - 8m + 12\)
= \(m^{2} - 6m - 2m + 12\)
= \(m\underline{(m - 6)} - 2\underline{(m - 6)}\)
= \((m - 6)(m - 2))\)

Reubstituting the value of \(m\),
= \([x^{2} - x - 6]\:[x^{2} - x - 2]\)
= \([x^{2} - 3x + 2x - 6]\:[x^{2} - 2x + x - 2]\)
= \([x\underline{(x - 3)} + 2\underline{(x - 3)}]\:[x\underline{(x - 2)} + 1\underline{(x - 2)}]\)
= \((x - 3)(x + 2)(x - 2)(x + 1)\)

Solution:

 \((x - 5)^{2} - (5x - 25) - 24\)

= \(\bbox[yellow, 1pt, border: 2px solid red]{(x - 5)}^{2} - 5\bbox[yellow, 1pt, border: 2px solid red]{(x - 5)} - 24\)

Let, \((x - 5) = m\)
∴ The given polynomial
= \(m^{2} - 5m - 24\)
= \(m^{2} - 8m + 3m - 24\)
= \(m\underline{(m - 8)} + 3\underline{(m - 8)}\)
= \((m - 8)(m + 3))\)

Resubstituting the value of \(m\),
= \((x - 5 - 8)\:(x - 5 + 3)\)
= \((x - 13)(x - 2)\)

Solution:

\((\bbox[yellow, 1pt, border: 2px solid red]{{x^{2} - 6x}})^{2} - 8(\bbox[yellow, 1pt, border: 2px solid red]{{x^{2} - 6x}}) + 8 - 64\)

Let, \(x^{2} - 6x = m\)
∴ The given polynomial
= \(m^{2} - 8(m + 8) - 64\)
= \(m^{2} - 8m - 64 - 64\)
= \(m^{2} - 8m - 128\)
= \(m^{2} - 16m + 8m - 128\)
= \(m\underline{(m - 16)} + 8\underline{(m - 16)}\)
= \((m - 16)(m + 8))\)

Reubstituting the value of \(m\),
= \([x^{2} - 6x - 16]\:[x^{2} - 6x + 8]\)
= \([x^{2} - 8x + 2x - 16]\:[x^{2} - 4x - 2x + 8]\)
= \([x\underline{(x - 8)} + 2\underline{(x - 8)}]\:[x\underline{(x - 4)} - 2\underline{(x - 4)}]\)
= \((x - 8)(x + 2)(x - 4)(x - 2)\)

Solution:

 \((\bbox[yellow, 1pt, border: 2px solid red]{{x^{2} - 2x}} + 3)(\bbox[yellow, 1pt, border: 2px solid red]{{x^{2} - 2x}} + 5) - 35\)

∴ The given polynomial
= \((m + 3)(m + 5) - 35\)
= \(m(m + 5) + 3(m + 5) - 35\)
= \(m^{2} + 5m + 3m + 15 - 35\)
= \(m^{2} + 8m - 20\)
= \(m\underline{(m + 10)} - 2\underline{(m + 10)}\)
= \((m + 10)(m - 2)\)

Reubstituting the value of \(m\),
= \((x^{2} - 2x + 10)\:(x^{2} - 2x - 2)\)

Solution:

\((y + 2)(y - 3)(y + 8)(y + 3) + 56\)
Let’s rearrange the brackets as shown:
 \(\bbox[yellow, 3pt, border: 2px dotted red]{(y + 2)(y + 3)}\ \bbox[pink, 3pt, border: 2px dotted red]{(y - 3)(y + 8)} + 56\)
= \(\bbox[yellow, 3pt, border: 2px dotted red]{[y(y + 3) + 2y(y + 3)]}\ \bbox[pink, 3pt, border: 2px dotted red]{[y(y + 8) - 3(y + 8)]} + 56\)
= \([y^{2} + 3y + 2y + 6]\:[y^{2} + 8y - 3y - 24] + 56\)
= \([\bbox[deepskyblue, 3pt, border: 2px dotted blue]{y^{2} + 5y} + 6]\:[\bbox[deepskyblue, 3pt, border: 2px dotted blue]{y^{2} + 5y} - 24] + 56\)

Now, let \(y^{2} + 5y = m\)
∴ The given polynomial
= \((m + 6)(m - 24) + 56\)
= \(m^{2} - 24m + 6m - 144 + 56\)
= \(m^{2} - 18m - 88\)
= \(m^{2} - 22m + 4m - 88\)
= \(m\underline{(m - 22)} + 4\underline{(m - 22)}\)
= \((m - 22)(m + 4)\)

Reubstituting the value of \(m\),
= \((y^{2} + 5y - 22)\:(y^{2} + 5y + 4)\)
= \((y^{2} + 5y - 22)\:[y^{2} + 4y + y + 4]\)
= \((y^{2} + 5y - 22)\:[y(y + 4) + 1(y + 4)]\)
= \((y^{2} + 5y - 22)\:(y + 4)(y + 1)\)

Solution:

\((\bbox[yellow, 5pt, border: 2px dotted red]{y^{2} + 5y})(\bbox[yellow, 5pt, border: 2px dotted red]{y^{2} + 5y} - 2) - 24\)

Now, let \(y^{2} + 5y = m\)
∴ The given polynomial
= \(m(m - 2) - 24\)
= \(m^{2} - 24m + 6m - 144 + 56\)
= \(m^{2} - 2m - 24\)
= \(m^{2} - 6m + 4m - 24\)
= \(m\underline{(m - 6)} + 4\underline{(m - 6)}\)
= \((m - 6)(m + 4)\)

Reubstituting the value of \(m\),
= \((y^{2} + 5y - 6)\:(y^{2} + 5y + 4)\)
= \([y^{2} + 6y - y - 6]\:[y^{2} + 4y + y + 4]\)
= \([y(y + 6) - 1(y + 6)]\:[y(y + 4) + 1(y + 4)]\)
= \((y + 6)(y - 1)(y + 4)(y + 1)\)

Solution:

\((x - 3)(x - 4)^{2}(x - 5) - 6\)
Let’s rearrange the brackets as shown
 \((x - 3)(x - 5)(x - 4)(x - 4) - 6\)
= \([x(x - 5) - 3(x - 5)]\:[x(x - 4) - 4(x - 4)] - 6\)
= \([x^{2} - 5x - 3x + 15][x^{2} - 4x - 4x + 16] - 6\)
= \([x^{2} - 8x + 15][x^{2} - 8x + 16] - 6\)

Now, let \(x^{2} - 8x = m\)
∴ The given polynomial
= \((m + 15)(m + 16) - 6\)
= \(m^{2} + 16m + 15m + 240 - 6\)
= \(m^{2} + 31m + 234\)
= \(m^{2} + 18m + 13m + 234\)
= \(m(m + 18) + 13(m + 18)\)

Reubstituting the value of \(m\),
= \((x^{2} - 8y + 18)\:(x^{2} - 8x + 13)\)