### Theory:

Let us expand some of the squared terms using the suitable standard identities.
1. $$(2x+3y)^2$$.

Let us use the identity, $$(a+b)^2$$ $$=$$ $$a^2+2ab+b^2$$.

Comparing $$(2x+3y)^2$$ with $$(a+b)^2$$, we have $$a=2x$$ and $$b=3y$$.

Substitute the values in the formula.

$$(2x+3y)^2$$ $$=$$ $$(2x)^2+(3y)^2+2(2x)(3y)$$

$$(2x+3y)^2$$ $$=$$ $$4x^2+9y^2+12xy$$.

2. $$(5x-7y)^2$$.

Let us use the identity, $$(a-b)^2$$ $$=$$ $$a^2-2ab+b^2$$.

Comparing $$(5x-7y)^2$$ with $$(a-b)^2$$, we have $$a=5x$$ and $$b=7y$$.

Substitute the values in the formula.

$$(5-7y)^2$$ $$=$$ $$(5x)^2+ (7y)^2-2(5x)(7y)$$

$$(5-7y)^2$$ $$=$$ $$25x^2+49y^2-70xy$$.

3. $$(x+5y)(x-5y)$$.

Let us use the identity, $$(a+b)(a-b)$$ $$=$$ $$a^2-b^2$$.

Comparing $$(x+5y)(x-5y)$$ with $$(a+b)(a-b)$$, we have $$a=x$$ and $$b=5y$$.

Substitute the values in the formula.

$$(x+5y)(x-5y)$$ $$=$$$$(x)^2-(5y)^2$$

$$(x+5y)(x-5y)$$ $$=$$$$x^2-25y^2$$.

4. $$(4y+5)(4y+3)$$.

Let us use the identity, $$(x+a)(x+b)$$ $$=$$ $$x^2+(a+b)x+ab$$.

Comparing $$(4y+5)(4y+3)$$ with $$(x+a)(x+b)$$, we have $$x=4y$$, $$a=5$$ and $$b=3$$.

Substitute the values in the formula.

$$(4y+5)(4y+3)$$ $$=$$ $$(4y)^2+(5+3)(4y)+(5)(3)$$

$$(4y+5)(4y+3)$$ $$=$$ $$16y^2+32y+15$$.

Example:
Look for the following cases where we used the identities.

1. Expand $$(x+4)^2$$ using the identity.

The above expression is in $$(a+b)^2$$ form.

We have the identity, $$(a+b)^2$$ $$=$$ $$a^2+2ab+b^2$$.

Substitute $$a = x$$ and $$b = 4$$ in the formula.

$$(x+4)^2$$ $$=$$ $$x^2+2(x)(4)+4^2$$

$$=$$ $$x^2+2\times 4x+16$$

$$=$$ $$x^2+8x+16$$

2. Evaluate $$98^2$$ using identity.

$$98^2$$ $$=$$ $$(100-2)^2$$

The above expression is in $$(a-b)^2$$ form.

We have the identity, $$(a-b)^2$$ $$=$$ $$a^2-2ab+b^2$$.

Substitute $$a = 100$$ and $$b = 2$$ in the formula.

$$(100-2)^2$$ $$=$$ $$100^2-2(100)(2)+2^2$$

$$= 10000-400+4$$

$$= 9604$$